1 Introduction
In this paper we study evolution, monotonicity, and differentiability of the first weighted p-eigenvalue on an n-dimensional compact Riemannian manifold \((M, g)\) equipped with measure dμ, whose metric \(g=g(t)\) evolves along the Ricci-harmonic flow (RHF)
Here \(\phi (x,t)=:\phi : M \times [0, \infty ) \to \mathbb{R}\) is a one-parameter family of smooth functions at least \(C^{2}\) in x and \(C^{1}\) in t, ⊗ is the tensor product, Rc is the Ricci curvature tensor of \((M,g)\), ∇ is the gradient operator, α is a nonincreasing constant function of time bounded below by \(\alpha _{n}>0\) in time, and Δ is the Laplace–Beltrami operator on M. Sometimes, we shall refer to (1.1) simply as RHF. System (1.1) was first studied by List [18] with motivation coming from general relativity. It was generalized by Müller [20] to the situation where \(\phi : (M,g) \to (N, h)\), (\((N, h)\) is a compact Riemannian manifold endowed with static metric h) and ϕ satisfies Eells and Sampson’s heat flow [9] for harmonic map. Indeed, if ϕ is a constant function, the flow degenerates to the well-known Hamilton’s Ricci flow [12].
$$ \textstyle\begin{cases} \frac{\partial }{\partial t} g(x ,t) = - 2 Rc (x ,t) + 2 \alpha \nabla \phi (x ,t) \otimes \nabla \phi (x ,t), \\ \frac{\partial }{\partial t} \phi (x ,t) = \Delta _{g} \phi (x ,t). \end{cases} $$
(1.1)
In recent time, getting useful information about behaviors of eigenvalues of geometric operators on evolving manifolds has gained more popularity among researchers. This information usually turns to powerful tools in the study of geometry and topology of the underlying manifolds. Perelman [21] recorded a great success by proving that the first nonzero eigenvalue of \(-4 \Delta + R \) is nondecreasing along the Ricci flow via the monotonicity formula for his energy functional \(\mathcal{F}\). Not quite long after Perelman’s paper [21], Cao [7] extended Perelman’s result to the first eigenvalue of \(\Delta + \frac{R}{2}\) on the condition that the curvature operator is nonnegative. Later, Li [15] proved the same result without any curvature assumption. The following papers [8, 16] are some results along this idea. Recently, the first author studied the evolution and monotonicity of the first eigenvalue of p-Laplacian and weighted Laplacian in [1] and [2], respectively. He found some monotonic quantities under the respective flows. Azami [5] (see also [6]) extended these results to the setting of Ricci–Bourguignon flow. In [11] and [10], the authors studied the evolution of the first eigenvalue of \(\Delta _{\phi }+ \frac{R}{2}\) under the Yamabe flow and the Ricci flow, respectively, and they also obtained some monotonic quantities under these flows. For similar results, see [3, 4, 17, 25, 26] and the references therein. Motivated by the above works, in this paper we extend the results in [1] and [2] to the first eigenvalue of weighted p-Laplacian under the Ricci-harmonic flow.
Anzeige
The plan of the paper is as follows. In Sect. 2, we give background information in terms of basic notation and relevant definitions. In Sect. 3 we discuss regularization procedure for the nonlinear and degenerate operator and then present some evolution equations that will be useful in the proofs of the main results. In the last section, we state and prove the main results of the paper. Here, we discuss the time evolution and monotonicity of \(\lambda _{1}(\Delta _{p,\phi })\) without differentiability assumption on its corresponding eigenfunction. In fact, the differentiability of the eigenvalue is a consequence of the monotonicity formula derived.
2 Notation and preliminaries
By standard notations in the theory of Ricci-harmonic flow, we denote a symmetric 2-tensor by \(Sc:= Rc-\alpha \nabla \phi \otimes \nabla \phi \), its components by \(S_{ij}:=R_{ij}-\alpha \phi _{i}\phi _{j}\), and its trace by \(S:=R-\alpha |\nabla \phi |^{2}\), where \(R_{ij}\) are the Ricci tensor’s components, R is the scalar curvature of \((M,g)\), and \(\phi _{i} = \nabla _{i}\phi = \frac{\partial }{\partial x^{i}} \phi \).
In most cases, our calculations will be performed in a local coordinate system \(\{x^{i}\}_{1}^{n}\), where repeated indices are summed. The Riemannian metric \(g(x)\) at any point \(x \in M\) is a bilinear symmetric positive definite matrix written in local coordinates as
We denote the Laplace–Beltrami operator, p-Laplacian, weighted Laplacian, and weighted p-Laplacian on \((M,g)\) by Δ, \(\Delta _{p}\), \(\Delta _{\phi }\), and \(\Delta _{p,\phi }\), respectively. For instance, in a local coordinate system
with respect to the Christoffel symbols \(\varGamma ^{k}_{ij}\), where \(g^{ij}=(g_{ij})^{-1}\) is the inverse matrix. We denote dv as the Riemannian volume measure on \((M,g)\) and \(d\mu :=e^{-\varphi (x)}\,dv\), the weighted volume measure, where \(\varphi \in C^{\infty }(M)\). Throughout, M will be assumed to be closed (compact without boundary), except if otherwise stated.
$$ g(x)=g_{ij} \,dx^{i}\,dx^{j}. $$
$$ \Delta = g^{ij} \biggl( \frac{\partial ^{2}}{\partial x^{i} \,\partial x^{j}} - \varGamma ^{k}_{ij} \frac{ \partial }{\partial x^{k}} \biggr) $$
Let \(f:M\to \mathbb{R}\) be a smooth function. The mini-max principle also holds for the weighted p-Laplacian where its first nonzero eigenvalue is characterized as follows:
satisfying the constraints \(\int _{M}|f|^{p-2} f \,d\mu =0\), where \(W^{1,p}(M,g,d\mu )\) is the completion of \(C^{\infty }(M,g,d\mu )\) with respect to the norm
The infimum in (2.1) is achieved by \(f \in W^{1,p}\) satisfying the Euler–Lagrange
for all \(\psi \in C^{\infty }_{0}(M)\) in the sense of distribution.
1.
For \(p \in (1,+\infty )\), the p-Laplacian of f is defined as
where div is the divergence operator, the adjoint of gradient for the \(L^{2}\)-norm induced by the metric on the space of differential forms. When \(p=2\), \(\Delta _{p}\) is the usual Laplace–Beltrami operator.
$$\begin{aligned} \Delta _{p} f & = \operatorname{div}\bigl( \vert \nabla f \vert ^{p-2} \nabla f\bigr) \\ & = \vert \nabla f \vert ^{p-2} \Delta f +(p-2) \vert \nabla f \vert ^{p-4} \operatorname{Hess} f ( \nabla f, \nabla f), \end{aligned}$$
2.
For the weighted volume measure \(d\mu = e^{-\phi }\,dv\), the weighted-Laplacian is defined by
which is a symmetric diffusion operator on \(L^{2}(M, g, d\mu )\) and self-adjoint with respect to the measure in the sense of integration by parts formula
for any \(u,v \in C^{\infty }(M)\) (M is closed). When ϕ is constant, the weighted Laplacian is just the Laplace–Beltrami operator.
$$ \Delta _{\phi }f:= e^{\phi }\operatorname{div}\bigl(e^{-\phi } \nabla f\bigr) = \Delta f - \langle \nabla \phi , \nabla f\rangle , $$
$$ \int _{M}\Delta _{\phi }u v \,d\mu = - \int _{M} \langle \nabla u, \nabla v \rangle \,d\mu = \int _{M} u \Delta _{\phi }v \,d\mu $$
3.
The weighted p-Laplacian generalizes the p-Laplacian and the weighted Laplacian. It is defined by
When \(p=2\), this is just the weighted Laplacian, and when ϕ is a constant, it is just the p-Laplacian.
$$ \Delta _{p,\phi } :=e^{\phi }\operatorname{div}\bigl(e^{-\phi } \vert \nabla f \vert ^{p-2} \nabla f\bigr) = \Delta _{p} f - \vert \nabla f \vert ^{p-2} \langle \nabla \phi , \nabla f\rangle . $$
$$\begin{aligned} \lambda _{1}(t) = \inf_{f} \biggl\{ \int _{M} \vert \nabla f \vert ^{p}\,d\mu : \int _{M} \vert f \vert ^{p}\,d\mu =1, f \neq 0, f \in W^{1,p}(M,g,d\mu ) \biggr\} \end{aligned}$$
(2.1)
$$ \Vert f \Vert _{W^{1,p}} = \biggl( \int _{M} \vert f \vert ^{p}\,d\mu + \int _{M} \vert \nabla f \vert ^{p} \,d \mu \biggr)^{\frac{1}{p}}. $$
$$\begin{aligned} \int _{M} \vert \nabla f \vert ^{p-2}\langle \nabla f, \nabla \psi \rangle \,d \mu -\lambda _{1} \int _{M} \vert f \vert ^{p-2}\langle f, \psi \rangle \,d\mu =0 \end{aligned}$$
(2.2)
Anzeige
We need to compute evolution of \(\lambda _{1}(t)\) but we know that it is nonlinear in general. Even we do not know whether \(\lambda _{1}(t)\) or its corresponding eigenfunction is \(C^{1}\)-differentiable along Ricci-harmonic flow. It has been pointed out that differentiability for the case \(p=2\) under geometric flow is a consequence of eigenvalue perturbation theory, see for instance [13]. To overcome this difficulty, we will use the approach of [7], see also Wu [25] and Wu, Wang, and Zheng [24], as used under the Ricci flow. The details will be discussed in Sect. 4.
3 Regularization procedure and basic lemma
Firstly, we introduce the linearized operator of the weighted p-Laplacian on function \(h \in C^{\infty }(M)\) defined pointwise at the points \(\nabla h \neq 0\) [23]
for a smooth function f̃ on M, where G can be viewed as a tensor defined as
Notice that \(\mathcal{L}_{\phi }\) is positive definite for \(p>1\) and strictly elliptic in general at these points (\(\nabla h \neq 0\)), and that the sum of its second order part is
with
When \(p \neq 2\), the weighted p-Laplacian degenerates or is singular at points \(\nabla f =0\). In this case ε-regularization technique is usually applied by replacing the linearized operator with its approximate operator, see [14, 22] for examples. For \(\varepsilon >0\), we define an approximate operator \(\mathcal{L}_{ \phi , \varepsilon } := \Delta _{p,\phi ,\varepsilon }\) for a smooth function \(f_{\varepsilon }\) by
with the following characterization:
where \(A_{\varepsilon }=|\nabla f_{\varepsilon }|^{2} + \varepsilon \).
$$\begin{aligned} \mathcal{L}_{\phi }(\tilde{f}) &:= e^{\phi }\operatorname{div} \bigl(e^{-\phi } \vert \nabla h \vert ^{p-2}G(\nabla \tilde{f}) \bigr) \\ &= \vert \nabla h \vert ^{p-2} \Delta _{\phi }\tilde{f} + (p-2) \vert \nabla h \vert ^{p-2} \operatorname{Hess} \tilde{f}(\nabla h, \nabla h) + (p-2)\Delta _{p,\phi }h \frac{ \langle \nabla h, \nabla \tilde{f}\rangle }{ \vert \nabla h \vert ^{2}} \\ &\quad{}+2(p-2) \vert \nabla h \vert ^{p-4}\operatorname{Hess} \tilde{f} \biggl( \nabla h, \nabla \tilde{f} - \frac{\nabla h}{ \vert \nabla h \vert } \biggl\langle \frac{\nabla h}{ \vert \nabla h \vert }, \nabla \tilde{f} \biggr\rangle \biggr) \end{aligned}$$
$$ G:=\mathrm{Id} + (p-2) \frac{\nabla h \otimes \nabla h}{ \vert \nabla h \vert ^{2}}. $$
$$ \mathbb{L}_{\phi }\tilde{f} := \vert \nabla h \vert ^{p-2} \Delta _{\phi } \tilde{f} + (p-2) \vert \nabla h \vert ^{p-2} \operatorname{Hess} \tilde{f}(\nabla h, \nabla h) $$
$$ \mathbb{L}_{\phi }h = \Delta _{p,\phi } h. $$
$$ \Delta _{p,\phi ,\varepsilon } f_{\varepsilon }= e^{\phi }\operatorname{div}\bigl(e ^{-\phi } A_{\varepsilon }^{\frac{p-2}{2}} \nabla f_{\varepsilon }\bigr) $$
$$ \lambda _{\varepsilon }= \inf_{f} \biggl\{ \int _{M} A_{\varepsilon } ^{\frac{p}{2}} \,d\mu : \int _{M} \vert f_{\varepsilon } \vert ^{p} \,d \mu =1, \int _{M} \vert f_{\varepsilon } \vert ^{p-2} f_{\varepsilon }\,d\mu =0 \biggr\} , $$
It has been shown in [22] that the infimum above is achieved by a \(C^{\infty }\) eigenfunction \(f_{\varepsilon }\) satisfying
with \(\lambda _{\varepsilon }= \int _{M} A_{\varepsilon }^{ \frac{p-2}{2}} |\nabla f_{\varepsilon }|^{2} \,d\mu \) by using standard elliptic theory. Taking the limit as \(\varepsilon \searrow 0\), we then obtain a continuous weak solution \(\lambda _{1} = \lim_{\varepsilon \searrow 0} \lambda _{\varepsilon }\) and \(f = \lim_{\varepsilon \searrow 0} f_{\varepsilon }\).
$$ \Delta _{p,\phi ,\varepsilon } f_{\varepsilon }= - \lambda _{\varepsilon } \vert f_{\varepsilon } \vert ^{p-2} f_{\varepsilon } $$
The following basic evolution formulas will be used in the proof of evolution of \(\lambda _{1}(t)\) under the Ricci-harmonic flow.
Lemma 3.1
Suppose that
\((M, g(t), \phi (t), d\mu )\), \(t \in [0,T]\), \(T<\infty \)
solves RHF (1.1). Then, for any
\(f \in C^{\infty }(M)\), we have the following formulas:
where
\(Z:= |\nabla f|^{p-2} \)
and
\(f_{t} = \frac{\partial }{\partial t}f\).
(1)
\(\frac{\partial }{\partial t} |\nabla f|^{p} = p|\nabla f|^{p-2} (S ^{ij}\nabla _{i} f\nabla _{j} f + g^{ij}\nabla _{i} f \nabla _{j} f_{t})\),
(2)
\(\frac{\partial }{\partial t} |\nabla f|^{p-2} = (p-2)|\nabla f|^{p-4} (S^{ij}\nabla _{i} f\nabla _{j} f + g^{ij}\nabla _{i} f \nabla _{j} f_{t})\),
(3)
\(\frac{\partial }{\partial t} (\Delta _{p} f) = 2 \mathcal{S}^{ij} \nabla _{i} ( Z \nabla _{j} f ) + g^{ij} \nabla _{i} ( Z_{t} \nabla _{j} f ) + g^{ij} \nabla _{i} (Z \nabla _{j} f_{t} ) - 2 \alpha Z (\Delta \phi ) g^{ij} \nabla _{i} \phi \nabla _{j} f\), and
(4)
$$\begin{aligned} \frac{\partial }{\partial t} (\Delta _{p, \phi } f) &= 2 \mathcal{S} ^{ij} \nabla _{i} ( Z \nabla _{j} f ) + g^{ij} \nabla _{i} ( Z_{t} \nabla _{j} f ) + g^{ij} \nabla _{i} (Z \nabla _{j} f_{t} ) \\ &\quad {}- 2 \alpha Z (\Delta \phi ) g^{ij} \nabla _{i} \phi \nabla _{j} f - Z _{t} \langle \nabla \phi , \nabla f \rangle - Z \langle \nabla \phi _{t}, \nabla f \rangle , \\ &\quad {} - Z \langle \nabla \phi , \nabla f_{t} \rangle - 2 S^{ij} Z \nabla _{i} \phi \nabla _{j} f, \end{aligned}$$
Proof
The proofs of formulas (1), (2), and (3) are contained in [1, Lemma 2.2]. For completeness, we sketch the proofs of (2) and (4) here.
Recall that \(\frac{\partial }{\partial t} g^{ij}= 2S^{ij}\) (see [1]) and \(A_{\varepsilon }:= |\nabla f_{\varepsilon }|^{2} + \varepsilon = g^{ij}\nabla _{i} f_{\varepsilon }\nabla _{j} f_{\varepsilon }+ \varepsilon \). Thus
Sending \(\varepsilon \searrow 0\), we arrive at formula (2). To prove formula (4), we write
Then
Note that \(\frac{\partial }{\partial t}(\Delta _{p,\varepsilon }f_{\varepsilon }) = \frac{\partial }{\partial t}(\Delta _{p}f)\) as \(\varepsilon \searrow 0\). This is formula (3). For the second term, we have
where \(Z_{\varepsilon }= A^{\frac{p-2}{2}}_{\varepsilon }\). Combining the computations and letting \(\varepsilon \searrow 0\), we arrive at the result. □
$$\begin{aligned} \frac{\partial }{\partial t} A^{\frac{p-2}{2}}_{\varepsilon } & = \frac{p-2}{2} A^{\frac{p-4}{2}}_{\varepsilon }\frac{\partial }{\partial t} A_{\varepsilon } \\ & = \frac{p-2}{2} A^{\frac{p-4}{2}}_{\varepsilon } \bigl(2S^{ij} \nabla _{i} f_{\varepsilon }\nabla _{j} f_{\varepsilon }+ 2g^{ij}\nabla _{i} f _{\varepsilon }\nabla _{j} (f_{\varepsilon })_{t} \bigr) \\ & = (p-2) A^{\frac{p-4}{2}} \bigl(S^{ij}\nabla _{i} f_{\varepsilon } \nabla _{j} f_{\varepsilon }+ g^{ij}\nabla _{i} f_{\varepsilon }\nabla _{j} (f_{\varepsilon })_{t} \bigr). \end{aligned}$$
$$ \Delta _{p,\phi , \varepsilon } f_{\varepsilon }= \Delta _{p,\varepsilon }f_{\varepsilon }- A^{\frac{p-2}{2}}_{\varepsilon }\langle \nabla \phi , \nabla f_{\varepsilon } \rangle . $$
$$\begin{aligned} \frac{\partial }{\partial t} (\Delta _{p,\phi , \varepsilon } f_{ \varepsilon }) = \frac{\partial }{\partial t}(\Delta _{p,\varepsilon }f _{\varepsilon }) - \frac{\partial }{\partial t} \bigl(A^{\frac{p-2}{2}} _{\varepsilon }g^{ij}\nabla _{i} \phi \nabla _{j} f_{\varepsilon } \bigr). \end{aligned}$$
$$\begin{aligned} \frac{\partial }{\partial t} \bigl(A^{\frac{p-2}{2}}_{\varepsilon }g ^{ij}\nabla _{i} \phi \nabla _{j} f_{\varepsilon } \bigr) &= \frac{\partial }{\partial t} \bigl(Z_{\varepsilon }g^{ij}\nabla _{i} \phi \nabla _{j} f _{\varepsilon } \bigr) \\ & = (Z_{\varepsilon })_{t} \langle \nabla \phi , \nabla f_{\varepsilon }\rangle + Z_{\varepsilon }\langle \nabla \phi _{t}, \nabla f_{\varepsilon }\rangle + Z_{\varepsilon }\bigl\langle \nabla \phi , \nabla (f_{\varepsilon })_{t} \bigr\rangle \\ &\quad{} + 2 S^{ij} Z_{\varepsilon }\nabla _{\phi }\nabla _{j} f_{\varepsilon }, \end{aligned}$$
4 Evolution of \(\lambda _{1}\) and monotonic quantities
In this section, we derive an evolution formula for the first nonzero eigenvalue of \(\Delta _{p,\phi }\) and show that \(\lambda _{1}\) is monotone nondecreasing along the Ricci-harmonic flow as a corollary. We also obtain some monotonic quantities involving \(\lambda _{1}\) which are also nondecreasing along the flow. In order to do these, we need to compute time derivatives of \(\lambda _{1}\) and its corresponding eigenfunction. Unfortunately, we do not know whether \(\lambda _{1}\) or its corresponding eigenfunction (\(p \neq 2\)) is \(C^{1}\) or not along the flow. As we remarked earlier, we adopt Cao’s approach [8] (see also [25] and [24]) to assume that \(\lambda _{1}(f(t),t) = \lambda _{1}(t)\) and that \(f(t)\) and \(\lambda _{1}(f(t),t)\) are smooth. Precisely, let \((M,g(t),\phi (t),d\mu )\), \(t \in [0,T]\) be a smooth compact solution of (1.1). Define a general smooth function as follows:
where \(f(t)\) is a smooth function satisfying the normalization condition
By this, we claim that there exists a smooth function \(f(t_{0})\) at time \(t=t_{0} \in [0,T]\) satisfying (4.1). To see this claim, we first assume that at \(t=t_{0}\), \(f(t_{0})\) is the eigenfunction corresponding to \(\lambda _{1}(t_{0})\) of \(\Delta _{p,\phi }\), which implies
Then we consider the following smooth function:
under the Ricci-harmonic flow \(g(t)\). We normalize this smooth function
under the flow \(g(t)\). By (4.4) we can easily check that \(f(t)\) satisfies (4.2). Note that in general \(\lambda _{1}(f,t)\) is not equal to \(\lambda _{1}(t)\). But at time \(t=t_{0}\), if \(f(t_{0})\) is the eigenfunction of the first eigenvalue \(\lambda _{1}(t _{0})\), then we conclude that
and that
at some time \(t=t_{0}\).
$$\begin{aligned} \lambda _{1}\bigl(f(t),t\bigr): = \int _{M} f(t) \Delta _{p,\phi } f(t) \,d\mu = \int _{M} \bigl\vert \nabla f(t) \bigr\vert ^{p} \,d\mu , \end{aligned}$$
(4.1)
$$\begin{aligned} \int _{M} \bigl\vert f(t) \bigr\vert ^{p}\,d\mu =1 \quad \text{and}\quad \int _{M} \bigl\vert f(t) \bigr\vert ^{p-2} f(t)\,d \mu =0. \end{aligned}$$
(4.2)
$$\begin{aligned} \int _{M} \bigl\vert f(t_{0}) \bigr\vert ^{p}\,d\mu =1 \quad \text{and}\quad \int _{M} \bigl\vert f(t_{0}) \bigr\vert ^{p-2} f(t_{0})\,d\mu =0. \end{aligned}$$
$$ u(t) = f(t_{0}) \biggl( \frac{\det (g(t_{0}))}{\det (g(t))} \biggr)^{ \frac{1}{2(p-2)}} $$
(4.3)
$$ f(t) = \frac{u(t)}{ ( \int _{M} \vert u(t) \vert ^{p} \,d\mu _{g(t)} )^{ \frac{1}{p}}} $$
(4.4)
$$ \lambda _{1}\bigl(f(t_{0}),t_{0}\bigr) = \lambda _{1}(t_{0}) $$
$$ \frac{d}{dt} \lambda _{1}\bigl(f(t_{0}), t_{0}\bigr)=\frac{d}{dt}\lambda _{1}(t _{0}) $$
We are now set to state the main results of this section.
Theorem 4.1
Let
\((M, g(t), \phi (t), d\mu )\), \(t \in [0,T]\), \(T<\infty \)
solve RHF (1.1) on a closed Riemannian manifold
M. Let
\(\lambda _{1}(t)\)
be the first nonzero eigenvalue of the weighted
p-Laplacian
\(\Delta _{p,\phi }\)
and
\(f(x,t)\)
its corresponding eigenfunction. Then
\(\lambda _{1}(t)\)
evolves by
for all time
\(t \in [0, T]\), \(T < \infty \).
$$ \begin{aligned}[b] \frac{d}{dt} \lambda _{1}(t) &= \lambda _{1}(t) \int _{M} (S+\phi _{t}) \vert f \vert ^{p} \,d\mu - \int _{M} (S+\phi _{t}) \vert \nabla f \vert ^{p} \,d\mu \\ &\quad {}+ p \int _{M} \vert \nabla f \vert ^{p-2} S^{ij} \nabla _{i} f \nabla _{j} f \,d\mu \end{aligned} $$
(4.5)
Corollary 4.2
Under the assumption of Theorem 4.1. Furthermore, if
\(S_{ij} \geq \beta (S+ \Delta \phi )g_{ij}\), \(\beta > \frac{1}{p}\)
is a constant. Then
where
for
\(t_{1} < t_{2}\), \(t_{1}, t_{2} \in [0,T]\), \(T< \infty \).
$$ \lambda _{1}(t_{2}) \geq \lambda _{1}(t_{1}) + \int _{t_{1}}^{t_{2}} \varTheta \bigl(g(t),f(x,t) \bigr) \,dt, $$
(4.6)
$$ \varTheta \bigl(g(t), f(x, t) \bigr) = \lambda _{1}(t) \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d\mu +(\beta p -1) \int _{M} (S+ \Delta \phi ) \vert \nabla f \vert ^{p} \,d \mu $$
Note that (4.5) in the theorem above is a general formula to describe the evolution of \(\lambda _{1}(t)\) under the Ricci-harmonic flow. Under some technical assumptions, we can obtain some monotonicity quantities.
Set \(S_{\mathrm{min}}(0) = \min_{x\in M} S(x,0)\), i.e., the minimum of \(S(x,t)\) with respect to \(g(t)\) and \(f(x,t)\) at \(t=0\).
Theorem 4.3
Let
\((M, g(t), \phi (t), d\mu )\), \(t \in [0,T]\), \(T<\infty \)
solve RHF (1.1) on a closed Riemannian manifold
M. Let
\(\lambda _{1}(t)\)
be the first nonzero eigenvalue of the weighted
p-Laplacian
\(\Delta _{p,\phi }\)
and
\(f(x,t)\)
its corresponding eigenfunction. Suppose
Furthermore, \(\lambda _{1}(t)\)
is differentiable almost everywhere along RHF (1.1).
(i)
\(S_{ij} \geq \beta (S+ \Delta \phi )g_{ij}\), \(\beta > \frac{1}{p}\)
is a constant;
(ii)
\(\Delta \phi \geq 0\), i.e., ϕ
is subharmonic;
(iii)
\(S \geq S_{\mathrm{min}}(0) \geq 0\), then
and
\(\lambda _{1}(t)\)
is monotonically nondecreasing along RHF (1.1);
$$\begin{aligned} \lambda _{1}(t_{2}) \geq \lambda _{1}(t_{1}) \exp \biggl(\beta p \int _{t _{1}}^{t_{2}} S_{\mathrm{min}}(t) \,dt \biggr) \end{aligned}$$
(4.7)
(iv)
If instead of (iii), \(S\geq S_{\mathrm{min}}(0) \neq 0\) (i.e., either
\(S_{\mathrm{min}}(0) >0\)
or
\(S_{\mathrm{min}}(0) <0\)), then
\(\lambda _{1}(t) (S_{\mathrm{min}}(0)^{-1} - \frac{2}{n}t )^{\frac{n\beta p}{2}}\)
is monotonically nondecreasing along RHF (1.1).
Corollary 4.4
Let
\((M, g(t), \phi (t), d\mu )\), \(t \in [0,T]\), \(T<\infty \)
solve RHF (1.1) on a compact Riemannian surface
\((M^{2}, g_{0})\). Let
\(\lambda _{1}(t)\)
be the first nonzero eigenvalue of
\(\Delta _{p,\phi }\)
with
\(f(x,t)\)
being its corresponding eigenfunction. Assume
ϕ
is subharmonic (i.e., \(\Delta \phi \geq 0\)).
(A)
Suppose
\(Rc \leq \epsilon \nabla \phi \otimes \nabla \phi \), where
\(\epsilon \leq \frac{2 \alpha (1-\beta )}{1-2\beta }\), \(\beta > \frac{1}{2}\).
(i)
If
\(S_{\mathrm{min}}(0) \geq 0\), then
\(\lambda _{1}(t)\)
is monotonically nondecreasing along RHF (1.1) for all
\(t \in [0,T]\), \(T<\infty \).
(ii)
If
\(S_{\mathrm{min}}(0) \neq 0\), then the quantity
\(\lambda _{1}(t) (S_{\mathrm{min}}(0)^{-1} - t )^{\beta p}\)
is monotonically nondecreasing along RHF (1.1) for all
\(t \in [0,T]\), \(T<\infty \).
(B)
Suppose that
\(\nabla \phi \otimes \nabla \phi \leq \frac{1}{2}|\nabla \phi |^{2} g _{ij}\).
(i)
If
\(S_{\mathrm{min}}(0) \geq 0\), then
\(\lambda _{1}(t)\)
is monotonically nondecreasing along RHF (1.1) for all
\(t \in [0,T]\), \(T<\infty \).
(ii)
If
\(S_{\mathrm{min}}(0) \neq 0\), then the quantity
\(\lambda _{1}(t) (S_{\mathrm{min}}(0)^{-1} - t )^{\beta p}\)
is monotonically nondecreasing along RHF (1.1) for all
\(t \in [0,T]\), \(T<\infty \).
Remark 4.5
Assuming the weight function \(\phi = \psi (x)\) is t-independent, i.e., \(d\mu = e^{-\psi (x)}\,dv\), then (4.5) of Theorem 4.1 becomes
Then, scaling \(\phi (x,t)\) by taking \(\phi = \sqrt{\frac{2}{\alpha }} u(x,t)\), RHF (1.1) becomes
studied by Li [17]. Thus, under these conditions our results in Theorem 4.1, Theorem 4.3, and Corollary 4.4 reduce to Theorem 1.5, Theorem 1.6, and Corollary 1.7 of [17], respectively.
$$ \frac{d}{dt} \lambda _{1}(t) = \lambda _{1}(t) \int _{M} S \vert f \vert ^{p} \,d \mu - \int _{M} S \vert \nabla f \vert ^{p} \,d\mu + p \int _{M} \vert \nabla f \vert ^{p-2} Sc( \nabla f, \nabla f) \,d\mu . $$
(4.8)
$$ \textstyle\begin{cases} \frac{\partial }{\partial t} g = - 2 Rc+ 4\nabla u \otimes \nabla u, \\ \frac{\partial }{\partial t} u = \Delta u \end{cases} $$
(4.9)
4.1 Proof of Theorem 4.1
Proof
The proof follows by direct computation using evolution formula (4) in Lemma 3.1. Let \(f(t_{0})\) and \(\lambda (t_{0}) = \lambda (f(t _{0}),t)\) be an eigenpair. Then, for a smooth function \(f(t)\), we can set
Then
Using evolution formula (4) of Lemma 3.1, we obtain
We now apply integration by parts formula on the first three terms on the right-hand side of (4.11). For the first term
The second term on the right-hand side of (4.12) can be written as (see the computation in Lemma A.1 below):
Substituting (4.13) into (4.12) we have
For the second term on the right-hand side of (4.11), integration by parts implies
Similarly, the third term on the right-hand side of (4.11) implies
Putting (4.14), (4.15), and (4.16) into (4.11), we obtain
Using the evolution formula (2) of Lemma 3.1 into (4.17) yields
A straightforward computation also yields
and
Putting (4.19) and (4.20) into (4.18) yields
Using the facts that
and the normalization condition \(\int _{M} |f|^{p} \,d\mu =1\), which implies
we arrive at
which is what we wanted to prove. □
$$ \lambda \bigl(f(t), t\bigr) = - \int _{M} f(t) \Delta _{p,\phi } f(t) \,d\mu . $$
$$\begin{aligned} \frac{d}{dt} \lambda _{1}(t)\bigg|_{t=t_{0}} = \frac{\partial }{\partial t}\lambda _{1}\bigl(f(t), t\bigr) = -\frac{\partial }{\partial t} \int _{M} f(t) \Delta _{p,\phi } f(t) \,d\mu . \end{aligned}$$
(4.10)
$$\begin{aligned} &\frac{\partial }{\partial t} \int _{M} f \Delta _{p,\phi } f \,d\mu \\ &\quad = \int _{M} \frac{\partial }{\partial t} ( \Delta _{p,\phi } f ) f \,d \mu + \int _{M} \Delta _{p,\phi } f \frac{\partial }{\partial t} ( f\,d \mu ) \\ &\quad = 2 \int _{M} \mathcal{S}^{ij} \nabla _{i} ( Z \nabla _{j} f )f \,d\mu + \int _{M} g^{ij} \nabla _{i} ( Z_{t} \nabla _{j} f ) f \,d\mu + \int _{M} g ^{ij} \nabla _{i} (Z \nabla _{j} f_{t} ) f \,d\mu \\ &\qquad {}- 2 \alpha \int _{M} Z (\Delta \phi ) g^{ij} \nabla _{i} \phi \nabla _{j} f f \,d\mu - \int _{M} Z_{t} \langle \nabla \phi , \nabla f \rangle f \,d\mu - \int _{M} Z \langle \nabla \phi _{t}, \nabla f \rangle f \,d\mu \\ &\qquad {}- \int _{M} Z \langle \nabla \phi , \nabla f_{t} \rangle f \,d\mu - 2 \int _{M} S^{ij} Z \nabla _{i} \phi \nabla _{j} f + \int _{M} \Delta _{p, \phi } f \frac{\partial }{\partial t} ( f\,d\mu ). \end{aligned}$$
(4.11)
$$ \begin{aligned}[b] 2 \int _{M} S^{ij} \nabla _{i} ( Z \nabla _{j} f) f \,d\mu &= - 2 \int _{M} Z \nabla _{i} f \nabla _{j} \bigl(S^{ij}f e^{-\phi } \bigr)\,dv = -2 \int _{M} Z S^{ij} \nabla _{i} f \nabla _{j} f \,d\mu \\ &\quad {}- 2 \int _{M} Z\nabla _{i} S^{ij} \nabla _{j} f f \,d\mu + 2 \int _{M} Z S ^{ij}\nabla _{i} f \nabla _{j} \phi f \,d\mu . \end{aligned} $$
(4.12)
$$\begin{aligned} - 2 \int _{M} Z\nabla _{i} S^{ij} \nabla _{j} f f \,d\mu & = \int _{M} S( \Delta _{p,\phi } f) f \,d\mu + \int _{M} S \vert \nabla f \vert ^{p} \,d\mu \\ &\quad {}+ 2 \alpha \int _{M} Z \Delta \phi \langle \nabla \phi , \nabla f \rangle \,d\mu . \end{aligned}$$
(4.13)
$$ \begin{aligned}[b] 2 \int _{M} S^{ij} \nabla _{i} ( Z \nabla _{j} f) f \,d\mu &= -2 \int _{M} Z S^{ij}\nabla _{i} f \nabla _{j} f \,d\mu + \int _{M} S(\Delta _{p,\phi } f) f \,d\mu \\ &\quad {}+ \int _{M} S \vert \nabla f \vert ^{p} \,d\mu+ 2 \alpha \int _{M} Z \Delta \phi \langle \nabla \phi , \nabla f \rangle \,d\mu\\&\quad{} + 2 \int _{M} Z S^{ij} \nabla _{i} f \nabla _{j} \phi f \,d\mu . \end{aligned} $$
(4.14)
$$ \begin{aligned}[b] \int _{M} g^{ij}\nabla _{i}(Z_{t} \nabla _{j} f) f\,d\mu &= - \int _{M} Z _{t} \nabla _{j} f \nabla ^{i}\bigl(fe^{-\phi }\bigr) \,dv \\ &= - \int _{M} Z_{t} \vert \nabla f \vert ^{2} \,d\mu + \int Z_{t} \langle \nabla f, \nabla \phi \rangle f \,d\mu . \end{aligned} $$
(4.15)
$$\begin{aligned} \int _{M} g^{ij} \nabla _{i}(Z\nabla _{j} f_{t}) f \,d\mu = - \int _{M} Z \langle \nabla f_{t}, \nabla f \rangle \,d\mu + \int _{m} Z \langle \nabla f_{t}, \nabla \phi \rangle f \,d\mu . \end{aligned}$$
(4.16)
$$ \begin{aligned}[b] \frac{\partial }{\partial t} \int _{M} f \Delta _{p,\phi } f \,d\mu &= -2 \int _{M} Z S^{ij}\nabla _{i} f \nabla _{j} f \,d\mu + \int _{M} S \Delta _{p,\phi } f f \,d\mu \\ &\quad {}+ \int _{M} S \vert \nabla f \vert ^{p} \,d\mu - \int _{M} Z_{t} \vert \nabla f \vert ^{2} \,d \mu - \int _{M} Z \langle \nabla f_{t}, \nabla f \rangle \,d\mu \\ &\quad {}- \int _{M} Z \langle \nabla \phi _{t}, \nabla f \rangle f \,d\mu + \int _{M} \Delta _{p,\phi } f \frac{\partial }{\partial t} ( f\,d\mu ). \end{aligned} $$
(4.17)
$$ \begin{aligned}[b] \frac{\partial }{\partial t} \int _{M} f \Delta _{p,\phi } f \,d\mu &= -p \int _{M} Z S^{ij}\nabla _{i} f \nabla _{j} f \,d\mu + \int _{M} S \Delta _{p,\phi } f f \,d\mu \\ &\quad {}+ \int _{M} S \vert \nabla f \vert ^{p} \,d\mu - (p-1) \int _{M} Z \langle \nabla f _{t}, \nabla f \rangle \,d\mu \\ &\quad {}- \int _{M} Z \langle \nabla \phi _{t}, \nabla f \rangle f \,d\mu + \int _{M} \Delta _{p,\phi } f \frac{\partial }{\partial t} ( f\,d\mu ). \end{aligned} $$
(4.18)
$$ \begin{aligned}[b] &- (p-1) \int _{M} Z \langle \nabla f_{t}, \nabla f \rangle \,d\mu \\ &\quad = (p-1) \int _{M} \nabla _{i}\bigl(Z\nabla _{j} f e^{-\phi }\bigr) f_{t} \,dv \\ &\quad = (p-1) \int _{M} \nabla _{i}(Z\nabla _{j} f)\,d\mu - (p-1) \int _{M} Z\langle \nabla f, \nabla \phi \rangle f_{t} \,d\mu \\ &\quad = (p-1) \int _{M}\Delta _{p,\phi } f f_{t} \,d\mu \end{aligned} $$
(4.19)
$$ \begin{aligned}[b] \int _{M} Z \langle \nabla \phi _{t}, \nabla f \rangle f \,d\mu &= - \int _{M} \phi _{t} \nabla _{i}\bigl(Z \nabla _{j} f f e^{-\phi }\bigr) \,dv \\ &= - \int _{M} \phi _{t} \Delta _{p,\phi } f \,d\mu - \int _{M} \phi _{t} \vert \nabla f \vert ^{p} \,d\mu . \end{aligned} $$
(4.20)
$$\begin{aligned} &\frac{\partial }{\partial t} \int _{M} f \Delta _{p,\phi } f \,d\mu \\ &\quad = -p \int _{M} Z S^{ij}\nabla _{i} f \nabla _{j} f \,d\mu + \int _{M} S \Delta _{p,\phi } f f \,d\mu \\ &\qquad{} + \int _{M} S \vert \nabla f \vert ^{p} \,d\mu + (p-1) \int _{M} \Delta _{p,\phi } f f_{t} \,d\mu + \int _{M} \phi _{t} \Delta _{p,\phi } f f \,d\mu \\ &\qquad{} + \int _{M} \phi _{t} \vert \nabla f \vert ^{p} \,d\mu + \int _{M} \Delta _{p,\phi } f \frac{\partial }{\partial t} ( f\,d\mu ) \\ & \quad = -p \int _{M} Z S^{ij}\nabla _{i} f \nabla _{j} f \,d\mu + \int _{M} S \Delta _{p,\phi } f f \,d\mu + \int _{M} S \vert \nabla f \vert ^{p} \,d\mu \\ &\qquad{} + \int _{M} \phi _{t} \Delta _{p,\phi } f f \,d\mu + \int _{M} \phi _{t} \vert \nabla f \vert ^{p} \,d\mu + \int _{M} \Delta _{p,\phi } f \biggl((p-1) f_{t} \,d \mu - \frac{\partial }{\partial t} ( f\,d\mu ) \biggr). \end{aligned}$$
$$\begin{aligned}& \frac{\partial }{\partial t} \int _{M} f \Delta _{p,\phi } f \,d\mu = - \frac{ \partial }{\partial t} \lambda _{1} \bigl(f(t),t\bigr)\bigg|_{t=t_{0}}, \\& \Delta _{p,\phi } f = - \lambda _{1}(t_{0}) \vert f \vert ^{p-2} f \end{aligned}$$
$$\begin{aligned} 0 &= \frac{\partial }{\partial t} \biggl( \int _{M} \vert f \vert ^{p} \,d \mu \biggr) = \frac{\partial }{\partial t} \biggl( \int _{M} \vert f \vert ^{p-1} f \,d \mu \biggr) \\ & = \int _{M} \vert f \vert ^{p-2} f \biggl[(p-1) f_{t} \,d\mu + \vert f \vert ^{p-1} \frac{ \partial }{\partial t} (f \,d \mu ) \biggr], \end{aligned}$$
$$\begin{aligned} \frac{d}{dt} \lambda _{1}\bigl(f(t),t\bigr)\bigg|_{t=t_{0}} & = \lambda _{1}(t_{0}) \int _{M} S \vert f \vert ^{p} \,d\mu - \int _{M} S \vert \nabla f \vert ^{p} \,d\mu + \lambda _{1}(t _{0}) \int _{M} \phi _{t} \vert f \vert ^{p} \,d\mu \\ & \quad{}- \int _{M} \phi _{t} \vert \nabla f \vert ^{p} \,d\mu + p \int _{M} \vert \nabla f \vert ^{p-2} S^{ij} \nabla _{i} f \nabla _{j} f \,d\mu , \end{aligned}$$
4.2 Proof of Corollary 4.2
Proof
Using the condition \(S_{ij} \geq \beta (S+\Delta \phi )g_{ij}\) in (4.5) of Theorem 4.1, we have
Denote
Integrating (4.21) from \(t_{1}\) to \(t_{2}\), \(t_{1} < t_{2}\), where \(t_{1}, t_{2} \in [0,T]\), \(T <\infty \) yields
which is the desired result. □
$$\begin{aligned} \frac{d}{dt} \lambda _{1}(t) \geq \lambda _{1}(t) \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d\mu +(\beta p -1) \int _{M} (S+ \Delta \phi ) \vert \nabla f \vert ^{p} \,d\mu . \end{aligned}$$
(4.21)
$$\begin{aligned} \varTheta \bigl(g(t), f(x, t) \bigr) := \lambda _{1}(t) \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d\mu +(\beta p -1) \int _{M} (S+ \Delta \phi ) \vert \nabla f \vert ^{p} \,d \mu . \end{aligned}$$
(4.22)
$$\begin{aligned} \lambda _{1}(t_{2})- \lambda _{1}(t_{1}) \geq \int _{t_{1}}^{t_{2}} \varTheta \bigl(g(t),f(x,t) \bigr) \,dt, \end{aligned}$$
4.3 Proof of Theorem 4.3
Proof
For \(t_{1} < t_{2}\), \(t_{1}, t_{2} \in [0,T]\), \(T< \infty \), for all time \(t \in [0, T]\), \(T < \infty \), we use (4.22)
Using the conditions \(\Delta \phi \geq 0\), \(S \geq S_{\mathrm{min}}(t)\) and \(\int _{M} |f|^{p} \,d\mu =1\), we have
Integrating between \(t_{1}\) and \(t_{2}\), \(t_{1}, t_{2} \in [0, T]\), we have
which yields (4.7).
$$\begin{aligned} &\varTheta \bigl(g(t), f(x, t) \bigr) \\ &\quad = \lambda _{1}(t) \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d\mu +(\beta p -1) \int _{M} (S+ \Delta \phi ) \vert \nabla f \vert ^{p} \,d \mu \\ &\quad = \lambda _{1}(t) \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d\mu +(\beta p -1) \int _{M} (S+ \Delta \phi ) \vert \nabla f \vert ^{p-2} g^{ij} \nabla _{i} f \nabla _{j} f \,d \mu \\ &\quad = \lambda _{1}(t) \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d\mu -(\beta p -1) \int _{M} (S+ \Delta \phi )\operatorname{div}\bigl(e^{-\phi } \vert \nabla f \vert ^{p-2} \nabla f\bigr)f \,dv \\ &\quad = \lambda _{1}(t) \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d\mu -(\beta p -1) \int _{M} (S+ \Delta \phi )\Delta _{p,\phi }f \cdot f \,d \mu \\ &\quad = \lambda _{1}(t) \beta p \int _{M} (S+ \Delta \phi ) \vert f \vert ^{p} \,d \mu . \end{aligned}$$
$$\begin{aligned} \frac{d}{dt} \lambda _{1}(t) \geq \lambda _{1}(t) \beta p S_{\mathrm{min}}(t). \end{aligned}$$
(4.23)
$$\begin{aligned} \ln \lambda _{1}(t_{2}) \geq \ln \lambda _{1}(t_{1}) + \beta p \int _{t _{1}}^{t_{2}} S_{\mathrm{min}}(t) \,dt, \end{aligned}$$
Now, set \(S_{\mathrm{min}}(0) = z_{0} \neq 0\). Recall that S satisfies an evolution equation [18, 20]
and inequality \(|S_{ij}|^{2} \geq \frac{1}{n} S^{2}\). Then solving
by applying the maximum principle, where one can compare S with the solution of an ODE \(z' = \frac{2}{n}z^{2}\), \(z(0)=z_{0} = S_{\mathrm{min}}(0)\), we have
Applying (4.23), we obtain
which implies
It is easy to check by elementary calculus that
Substituting (4.25) into (4.24), we obtain
for any time \(t_{1} < t_{2}\). By this we have
which means that the quantity \(\lambda _{1}(t) (z_{0}^{-1} - \frac{2}{n}t )^{\frac{n}{2}\beta p}\) is nondecreasing in the interval \([t_{1}, t_{2}]\) along the Ricci-harmonic flow. Notice that \((z_{0}^{-1} - \frac{2}{n}t)\) is decreasing in the interval \(t_{1}< t_{2}\), \(t_{1}, t_{2} \in [0, T)\). This means that \(\lambda _{1}(t)\) is nondecreasing along the flow. This completes the proofs of monotonicity.
$$ \frac{\partial S}{\partial t} = \Delta S + 2 \vert S_{ij} \vert ^{2} + 2 \alpha \vert \Delta \phi \vert ^{2} $$
$$ \frac{\partial S}{\partial t} \geq \Delta S + \frac{2}{n} \vert S_{ij} \vert ^{2} $$
$$ S(t) \geq z(t) = \frac{1}{z_{0}^{-1}-\frac{2}{n} t}, \quad t \in [0, T], T < \frac{n}{2 S_{\mathrm{min}}(0)} < \infty . $$
$$\begin{aligned} \frac{d}{dt} \lambda _{1}(t) \geq \lambda _{1}(t) \beta p \cdot \frac{1}{z _{0}^{-1}-\frac{2}{n} t}, \end{aligned}$$
$$\begin{aligned} \lambda _{1}(t_{2}) \geq \lambda _{1}(t_{1}) \exp \biggl(\beta p \int _{t _{1}}^{t_{2}} \frac{dt}{z_{0}^{-1}-\frac{2}{n} t} \biggr). \end{aligned}$$
(4.24)
$$\begin{aligned} \int _{t_{1}}^{t_{2}} \frac{dt}{z_{0}^{-1}-\frac{2}{n} t} = \ln \biggl( \frac{z _{0}^{-1} - \frac{2}{n}t_{1}}{z_{0}^{-1} - \frac{2}{n}t_{2}} \biggr)^{ \frac{n}{2}}. \end{aligned}$$
(4.25)
$$\begin{aligned} \ln \biggl(\frac{\lambda _{1}(t_{2})}{\lambda _{1}(t_{1})} \biggr) \geq \ln \biggl(\frac{z_{0}^{-1} - \frac{2}{n}t_{1}}{z_{0}^{-1} - \frac{2}{n}t_{2}} \biggr)^{\frac{n}{2}\beta p} \end{aligned}$$
$$\begin{aligned} \lambda _{1}(t_{2}) \biggl(z_{0}^{-1} - \frac{2}{n}t_{2} \biggr)^{ \frac{n}{2}\beta p} \geq \lambda _{1}(t_{1}) \biggl(z_{0}^{-1} - \frac{2}{n}t_{1} \biggr)^{\frac{n}{2}\beta p}, \end{aligned}$$
For differentiability of \(\lambda _{1}(t)\), it is easy to see that \(\lambda _{1}(t)\) is differentiable almost everywhere by the classical Lebesgue’s theorem [19, Chap. 4] since \(\lambda _{1}(t)\) is nondecreasing on the time interval \([0,T)\). □
Remark 4.6
By the maximum principle, the assumption that ϕ is subharmonic implies that the maximum of ϕ cannot be achieved in M since ϕ is not a constant function; however, its minimum can be achieved in M.
Remark 4.7
Our proofs of evolution and monotonicity of the first eigenvalue of \(\Delta _{p,\phi }\) do not use any differentiability of the first eigenvalue or its corresponding eigenfunction under the Ricci-harmonic flow. In fact, it is not known whether they are differentiable in advance.
4.4 Proof of Corollary 4.4
Proof
Note that on \(M^{2}\) we have \(R_{ij} = \frac{1}{2} R g_{ij}\). Then we compute
For any nonzero vector \(X =(X^{i})\) and the condition \(R_{ij} \leq \epsilon \nabla _{i} \phi \nabla _{j} \phi \), we have
From here one can conclude that \([ (\frac{1}{2}-\beta ) \epsilon -(1-\beta )\alpha ] |\nabla \phi |^{2} \geq 0\) since \(\epsilon \leq \frac{2\alpha (1-\beta )}{1-2\beta }\), \(\beta > \frac{1}{2}\), which implies that the condition \(S_{ij}-\beta Sg_{ij} \geq 0\), \(\beta > \frac{1}{2}\) holds on \(M^{2}\). Then consequences \(A(i)\) and \(A(ii)\) of the corollary follow from Theorem 4.3.
$$\begin{aligned} S_{ij} -\beta S g_{ij} & = \frac{R}{2} g_{ij} - \alpha \nabla _{i} \phi \nabla _{j} \phi - \beta \bigl(R- \alpha \vert \nabla \phi \vert ^{2}\bigr)g_{ij} \\ & = \biggl(\frac{1}{2}-\beta \biggr)Rg_{ij} - \alpha \nabla _{i} \phi \nabla _{j} \phi - \beta \alpha \vert \nabla \phi \vert ^{2}g_{ij}. \end{aligned}$$
$$\begin{aligned} (S_{ij} -\beta S g_{ij})X^{i} X^{j} & = \biggl(\frac{1}{2}-\beta \biggr) R \vert X \vert ^{2} - \alpha \langle \nabla \phi , X \rangle ^{2} + \beta \alpha \vert \nabla \phi \vert ^{2} \vert X \vert ^{2} \\ & \geq \biggl[ \biggl(\frac{1}{2}-\beta \biggr)\epsilon -(1-\beta )\alpha \biggr] \vert \nabla \phi \vert ^{2} \vert X \vert ^{2}. \end{aligned}$$
On the other hand, we can show that \(\nabla \phi \otimes \nabla \phi \leq \frac{1}{2} |\nabla \phi |^{2} g_{ij}\) holds on the Riemann surface \(M^{2}\). Note that
meaning that
which implies \(S_{ij}-\frac{1}{2}S g_{ij} \geq 0\). Then consequences \(B(i)\) and \(B(ii)\) follow from Theorem 4.3. □
$$\begin{aligned} (S_{ij} -\beta S g_{ij}) X^{i} X^{j} & = \biggl[\frac{R}{2} g_{ij} - \alpha \nabla _{i} \phi \nabla _{j} \phi - \beta \bigl(R- \alpha \vert \nabla \phi \vert ^{2}\bigr)g_{ij} \biggr] X^{i} X^{j} \\ & = \frac{R}{2} \vert X \vert ^{2} - \alpha \langle \nabla \phi , X \rangle ^{2} - \beta \bigl(R- \alpha \vert \nabla \phi \vert ^{2}\bigr) \vert X \vert ^{2} \\ & \geq \frac{R}{2} \vert X \vert ^{2} - \frac{\alpha }{2} | \vert \nabla \phi \vert ^{2} \vert X \vert ^{2} - \beta \bigl(R- \alpha \vert \nabla \phi \vert ^{2}\bigr) \vert X \vert ^{2} \end{aligned}$$
$$ S_{ij}X^{i} X^{j} \geq \frac{1}{2} \bigl(R- \alpha \vert \nabla \phi \vert ^{2} \bigr) \vert X \vert ^{2}, $$
Acknowledgements
The authors wish to thank the two anonymous referees and the editor for their useful comments.
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