Geometric Analysis

Frontmatter

Big and Nef Classes, Futaki Invariant and Resolutions of Cubic Threefolds

Abstract

In this note we revisit and extend few classical and recent results on the definition and use of the Futaki invariant in connection with the existence problem for Kähler constant scalar curvature metrics on polarized algebraic manifolds, especially in the case of resolution of singularities. The general inspiration behind this work is no doubt the beautiful paper by Ding and Tian [16] which contains the germs of a huge amount of the successive developments in this fundamental problem, and it is a great pleasure to dedicate this to Professor G. Tian on the occasion of his birthday.

Claudio Arezzo, Alberto Della Vedova

Bottom of Spectra and Amenability of Coverings

Abstract

For a Riemannian covering π : M1 → M0, the bottoms of the spectra of M0 and M1 coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of M0.

Werner Ballmann, Henrik Matthiesen, Panagiotis Polymerakis

Some Remarks on the Geometry of a Class of Locally Conformally Flat Metrics

Abstract

We prove that conformal metrics on domains of the round sphere, with non-negative constant Q-curvature, and non-negative scalar curvature, has positive mean curvature on the boundary of embedded balls (in the round metric). As a result, such metrics have certain reflection symmetries if the complement of the domain is contained in a lower-dimensional round sphere. We also prove that the development map of a locally conformally flat metric with non-positive Schouten tensor is an embedding.

Sun-Yung A. Chang, Zheng-Chao Han, Paul Yang

Analytical Properties for Degenerate Equations

Abstract

By a classical result, solutions of analytic elliptic PDEs, like the Laplace equation, are analytic. In many instances, the properties that come from being analytic are more important than analyticity itself. Many important equations are degenerate elliptic and solutions have much lower regularity. Still, one may hope that solutions share properties of analytic functions. These properties are closely connected to important open problems. In this survey, we will explain why solutions of an important degenerate elliptic equation have analytic properties even though the solutions are not even C3.

Tobias Holck Colding, William P. Minicozzi II

Equivariant K-theory and Resolution I: Abelian Actions

Abstract

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type, in an iterated sense, with connecting fibrations and this structure descends to a resolution of the quotient. For an Abelian group action the equivariant K-theory can then be described in terms of bundles over the base with morphisms covering the connecting maps. A similar model is given, in terms of appropriately twisted deRham forms over the base as an iterated space, for delocalized equivariant cohomology in the sense of Baum, Brylinski and MacPherson. This approach allows a direct proof of their equivariant version of the Atiyah–Hirzebruch isomorphism.

Panagiotis Dimakis, Richard Melrose

On the Existence Problem of Einstein–Maxwell Kähler Metrics

Abstract

In this expository paper we review on the existence problem of Einstein–Maxwell Kähler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein–Maxwell Kähler metrics, and give extensions of volume minimization principle, the notion of toric Kstability and other related results to the general set-up. Secondly, we consider the toric case when the manifold is the one point blow-up of the complex project plane and the Kähler class Ω is chosen so that the area of the exceptional curve is sufficiently close to the area of the rational curve of selfintersection number 1. We observe by numerical analysis that there should be a Killing vector field K which gives a toric K-stable pair (Ω,K) in the sense of Apostolov–Maschler.

Akito Futaki, Hajime Ono

Local Moduli of Scalar-flat Kähler ALE Surfaces

Abstract

In this article, we give a survey of our construction of a local moduli space of scalar-flat Kähler ALE metrics in complex dimension 2.We also prove an explicit formula for the dimension of this moduli space on a scalar-flat Kähler ALE surface which deforms to the minimal resolution of C2/Γ, where Γ is a finite subgroup of U(2) without complex reflections, in terms of the embedding dimension of the singularity.

Jiyuan Han, Jeff A. Viaclovsky

Singular Ricci Flows II

Abstract

We establish several quantitative results about singular Ricci flows, including estimates on the curvature and volume, and the set of singular times.

Bruce Kleiner, John Lott

An Inequality Between Complex Hessian Measures of Hölder Continuous m-subharmonic Functions and Capacity

Abstract

For a Riemannian covering π : M1 → M0, the bottoms of the spectra of M0 and M1 coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of M0.

Sławomir Kołodziej, Ngoc Cuong Nguyen

A Guided Tour to Normalized Volume

Abstract

This is a survey on the recent theory on minimizing the normalized volume function attached to any klt singularities.

Chi Li, Yuchen Liu, Chenyang Xu

Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry

Abstract

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is C1,1. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by C1,1 solutions on larger and larger compact domains, and, in particular, for entire C1,1 loc solutions: they are either constants or standard bubbles.

YanYan Li, Luc Nguyen, Bo Wang

Arsove–Huber Theorem in Higher Dimensions

Abstract

In this note we briefly present the progress in the research project to extend Huber’s theory of surfaces to general dimensions. The full paper [42] is in progress. We discuss n-Laplace equations and n-subharmonic functions using nonlinear potential theory. Particularly we build the Brezis–Merle type sharp inequality for Wolff potential and establish Taliaferro’s estimates in higher dimensions. We then apply the theory of n-subharmonic functions developed here to study hypersurfaces in hyperbolic space with nonnegative Ricci curvature as well as locally conformal flat manifolds with nonnegative Ricci.

Shiguang Ma, Jie Qing

From Local Index Theory to Bergman Kernel: A Heat Kernel Approach

Abstract

The aim of this note is to explain a uniform approach of three different topics: Atiyah–Singer index theorem, holomorphic Morse inequalities and asymptotic expansion of Bergman kernel, by using heat kernels.

Xiaonan Ma

Fourier–Mukai Transforms, Euler–Green Currents, and K-Stability

Abstract

Inspired by Gang Tian’s work in [4, 10, 11], and [12] we exhibit a wide range of energy functionals in Khler geometry as Fourier–Mukai transforms. Consequently these energies are completely determined by dual type varieties and therefore have logarithmic singularities when restricted to the space of algebraic potentials. This paper is dedicated to Gang Tian on the occasion of his 60th birthday.

Sean Timothy Paul, Kyriakos Sergiou

The Variations of Yang–Mills Lagrangian

Abstract

We are giving a survey on some of the analysis methods from gauge theory developed in the last decades. We first cover Uhlenbeck’ s compensated compactness theory in critical 4 dimension for the Yang–Mills functional. As an application we present the resolution of minimization processes of Yang– Mills in this critical dimension. In the second part of the survey we present the resolution of similar variational questions in super-critical dimensions and we end up the survey by stating some open problems raised by Tian relative to the regularity of Ω-anti-self-dual instantons in high dimensions.

Tristan Rivière

Tian’s Properness Conjectures:An Introduction to Kähler Geometry

Abstract

This manuscript served as lecture notes for a minicourse in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kähler geometry by describing the recent resolution of Tian’s three influential properness conjectures in joint work with T. Darvas. These results – inspired by and analogous to work on the Yamabe problem in conformal geometry – give an analytic characterization for the existence of Kähler–Einstein metrics and other important canonical metrics in complex geometry, as well as strong borderline Sobolev type inequalities referred to as the (strong) Moser–Trudinger inequalities.

Yanir A. Rubinstein

Ancient Solutions in Geometric Flows

Abstract

In this survey paper we discuss ancient solutions to different geometric flows, such as the Ricci flow, the mean curvature flow and the Yamabe flow. We survey the classification results of ancient solutions in the Ricci flow and the mean curvature flow. We also discuss methods for constructing new ancient solutions to the Yamabe flow, indicating that the classification results for this flow are impossible to expect.

Natasa Sesum

The Kähler–Ricci Flow on CP2

Abstract

We give a direct proof of the convergence of the Kähler–Ricci flow on CP2 without assuming the existence of the Kähler–Einstein metric.

Jian Song

Pluriclosed Flow and the Geometrization of Complex Surfaces

Abstract

We recall fundamental aspects of the pluriclosed flow equation and survey various existence and convergence results, and the various analytic techniques used to establish them. Building on this, we formulate a precise conjectural description of the long time behavior of the flow on complex surfaces. This suggests an attendant geometrization conjecture which has implications for the topology of complex surfaces and the classification of generalized Kähler structures.

Jeffrey Streets

From Optimal Transportation to Conformal Geometry

Abstract

In this paper we discuss the link between domain convexity in optimal transportation and the estimation of second derivatives in augmented Hessian equations, leading to the estimation of second derivatives in fully nonlinear Yamabe problems with boundary with boundary curvature conditions which may be also nonlinear.

Neil S. Trudinger

Special Lagrangian Equations

Abstract

We survey special Lagrangian equation and its related fully nonlinear elliptic and parabolic equations: definition, geometric background, basic properties, and progress. These include the rigidity of entire solutions, a priori Hessian estimates, construction of singular solutions, existence, the counterparts in the parabolic-curvature flow-settings, and open problems.

Yu Yuan

Positive Scalar Curvature on Foliations:The Enlargeability

Abstract

We generalize the famous result of Gromov and Lawson on the nonexistence of metric of positive scalar curvature on enlargeable manifolds to the case of foliations, without using index theorems on noncompact manifolds.

Weiping Zhang

Kähler–Einstein Metrics on Toric Manifolds and G-manifolds

Abstract

This is an expository paper. In the first part, we discuss variant approaches in the study of Kähler–Einstein metrics on toric Fano manifolds. In the second part, we discuss the existence problem of Kähler–Einstein metrics on G-manifolds via Mabuchi’s K-energy. Our method can be regarded as an extension in toric Fano manifolds. Some remaining questions/problems are also discussed.

Xiaohua Zhu

Some Questions in the Theory of Pseudoholomorphic Curves

Abstract

This survey article, in honor of G. Tian’s 60th birthday, is inspired by R. Pandharipande’s 2002 note highlighting research directions central to Gromov–Witten theory in algebraic geometry and by G. Tian’s complexgeometric perspective on pseudoholomorphic curves that lies behind many important developments in symplectic topology since the early 1990s.

Aleksey Zinger