Spectrum Estimates and Applications to Geometry

\(c(M)>0\) if M is compact with non-empty boundary.

Piecewise smooth boundary here means that there is a closed set \(Q\subset \partial M\) of \((n-1)\)-Hausdorff measure zero such that for each point \(q \in \partial M\setminus Q\) there is a neighborhood of q in \(\partial M\) that is a graph of a smooth function over the tangent space \(T_{q}\partial M\), see Whitney [115] pages 99–100.

Sometimes the mean curvature vector is defined as \(H(q) = \sum _{i=1}^{k}\mathcal S^\mathcal F(q)(e_{i}, e_{i})\).

In fact, a gradient is basic if and only if it is horizontal.

A continuous embedding \(\gamma :\mathbb {S}^{1} \rightarrow \mathbb {R}^{3}\).

Here, \(\sigma (M)\) for incomplete M is defined as the spectrum of the Friedrichs extension of \((-\Delta , C^\infty _c(M))\).

Denoting with \(\Delta _2\) the Laplace operator on \(L^2\), the semigroup \(e^{t\Delta _2}\) extends to a strongly continuous contraction semigroup \(T_p\) on \(L^p(M)\) for all \(p \in [1,+\infty )\). By definition, the \(L^p\)-spectrum is the spectrum of the generator \(\Delta _p\) of \(T_p\), and \(\Delta _\infty \) is the adjoint of \(\Delta _1\).

The volume condition was absent in the original formulation of the conjecture in [30], but was pointed out to us by the authors themselves after they discovered J. Lott’s paper.

Whether \(\Omega =\mathbb {D} \) or \(\Omega =\mathbb {A}(1/c,c))\).