# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Ambient metrics and exceptional holonomy

Construction of pseudo-Riemannian metrics whose holonomy is an exceptional Lie group has been of interest in recent years. This talk will outline a construction of an infinite-dimensional family of metrics in dimension 7 whose holonomy is contained in the split real form of the exceptional group $G_2$. An open dense subset of the family has holonomy equal to $G_2$. The datum for the construction is a generic real-analytic 2-plane field on a manifold of dimension 5; the metric in dimension 7 arises as the ambient metric of a conformal structure on the 5-manifold defined by Nurowski in terms of the 2-plane field.

This is work with Travis Willse and generalizes results of Leistner and Nurowski.

##### Improved Moser-Trudinger inequalities and Liouville equations on compact surfaces

We consider a class of equations with exponential nonlinearities and possibly singular sources motivated from the study of abelian Chern Simons models or from the problem of prescribing a metric with conical singularities through conformal transformations. Using new improvements of the classical Moser-Trudinger inequality and, combined with topological methods, we derive some general existence results.

##### Hessian estimates for special Lagrangian equations with critical and supercritical phases

We talk about a priori Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. The "gradient" graphs of solutions are minimal Lagrangian submanifolds. Our unified approach leads to sharper estimates even for the previously known three dimensional or convex solution cases. Recent counterexamples for subcritical phase equations will also be mentioned.

This is joint work with Dake Wang.

##### A Bernstein type theorem for entire Willmore graphs

We show that every two-dimensional entire graphical solution to the Willmore equation with square integrable second fundamental form is a plane. This is joint work with Tobias Lamm.

##### Obstruction-Flat Asymptotically Locally Euclidean Metrics

Given an even dimensional Riemannian manifold $(M^{n},g)$ with $n\ge 4$, it was shown in the work of Charles Fefferman and Robin Graham on conformal invariants the existence of a non-trivial 2-tensor which involves $n$ derivatives of the metric, arises as the first variation of a conformally invariant functional and vanishes for metrics that are conformally Einstein. This tensor is called the Ambient Obstruction tensor and is a higher dimensional generalization of the Bach tensor in dimension 4. We show that any asymptotically locally Euclidean (ALE) metric which is obstruction flat and scalar-flat must be ALE of a certain optimal order using a technique developed by Cheeger and Tian for Ricci-flat metrics. We also prove a singularity removal theorem for obstruction-flat metrics with isolated $C^{0}$-orbifold singularities.

##### Sharp constants in inequalities on the Heisenberg group

We derive the sharp constants for the inequalities on the Heisenberg group whose analogues on Euclidean space are the well known Hardy-Littlewood-Sobolev inequalities. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago, which was crucial in the solution of the CR Yamabe problem. Our methodology is completely diﬀerent from that used to obtain the Euclidean inequalities and can be used to give a new, rearrangement free, proof of the HLS inequalities. The talk is based on joint work with E. H. Lieb.

##### Curvature of random metrics

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics. This is joint work with I. Wigman and Y. Canzani

##### The shooting method and the analysis of the target map via the degree theory

We introduce and analysis the ‘target map’ for the shooting method. For a large class of elliptic systems as well as more general dynamic systems, we show that the target map is onto via the degree theory. The target map is onto implies that we can shoot to any desired position.

Applying our result to a motivating example, we obtain the existence of global positive solutions to the Hardy-Littlewood-Sobolev (also known as Lane-Emden) type system:

$$

\begin{cases}

& (-\Delta)^k u = v^p, \text{in $\mathbb{R}^n$},\\

& (-\Delta)^k u = v^q, \text{in $\mathbb{R}^n$},\\

\end{cases}

$$

##### $W^{2,1}$ regularity for the Monge-Ampère equation

The Monge-Ampère equation arises in connections with several problems from geometry and analysis (optimal transport, the Minkowski problem, the affine sphere problem, etc.) The regularity theory for this equation has been widely studied. In particular, in the 90's Caffarelli developed a regularity theory for Aleksandrov/viscosity solutions, showing that strictly convex solutions of $det(D^2u)=f$ are $C^{1,\alpha}$ provided $f$ is bounded away from zero and infinity, and are $W^{2,p}$ with $p>1$ if $f$ is uniformly close to a constant. The counterexamples by Wang in 1995 showed that the results of Caffarelli were more or less optimal. However, an important question which remained open was whether solutions with right hand side bounded away from zero and infinity could be $W^{2,1}$.

##### Large Time Behavior of Periodic Viscosity Solutions of Integro-differential Equations

In this talk, I will present some recent results on the asymptotic behavior of periodic viscosity solutions of parabolic integro-differential equations (where nonlocal terms are associated with Levy-It\^o operators). In particular, we address the problem to mixed integro-differential equations, e.g. where fractional diffusion occurs in one direction and classical diffusion in the complementary one. The typical results states that, under some suitable assumptions, the solution of the initial value problem behaves like $ct + v(x) + o(1)$ as $t\rightarrow\infty$, where $v$ is a solution of the stationary ergodic problem corresponding to the ergodic constant $c$.

##### H-Projective Geometry on Compact Kähler Manifolds

The basic geometric structure in h-projective geometry is the family of h-planar curves, associated to a given Kähler metric. Such curves can be seen as generalisations of geodesics on Kähler manifolds. In this context, one problem of interest is the investigation of Kähler manifolds admitting another Kähler metric having the same h-planar curves as the given one. Such a pair of metrics is called h-projectively equivalent. Besides a general introduction to h-projective geometry I want to present a result which was obtained in a joint work with V. S.

##### Loss of Compactness and Bubbling in the Space of Complete Minimal Surfaces in Hyperbloc Space

We consider the space of complete minimal surfaces in $H^3$ with a (free) boundary at infinity. We explain how the Willmore energy is a natural functional on this space. We study the possible loss of compactness in the space of such surfaces with energy bounded above. This question has been extensively studied for various energies in the context of closed surfaces, starting with the classical work of Sacks and Uhlenbeck on harmonic maps. We derive analogues of epsilon-reglarity, removability of singularities and bubbing in this setting. A key difference (and difficulty) compared to the classical picture is a lack of energy quantization. Joint with R. Mazzeo.

##### Bocher Type Theorems for Degenerate Conformally Invariant Equations

A classical theorem of Bocher states that a positive harmonic function in a punctured ball can be written as the sum of a multiple of the fundamental solution and a regular harmonic function. I will describe generalizations for degenerate $\sigma_k$ equations. Joint work with Yanyan Li.

##### The Spacetime Positive Mass Theorem in Dimensions Less Than 8

After reviewing the proof of the Riemannian positive mass theorem in dimensions less than 8, I will briefly explain how to generalize the proof to slices of spacetime that are not time- symmetric. The basic idea is to replace minimal hypersurfaces by marginally outer-trapped hypersurfaces, and the main difficulty is to avoid using any minimization process. This is joint work with Eichmair, Huang, and Schoen.

##### $L^p$ Bounds for Eigenfunctions on Locally Symmetric Spaces

There is a classical theorem of Sogge which provides bounds for the $L^p$ norms of a Laplace eigenfunction on a compact Riemannian manifold, which are sharp on the sphere and for spectral clusters. I will present a generalization of this theorem to eigenfunctions of the full ring of invariant differential operators on a locally symmetric space, as well as a theorem on the restriction of eigenfunctions to maximal flat subspaces. Time permitting, I will discuss ways in which these bounds can be improved using inputs from number theory.

##### Singular Metrics and the Calabi-Yau Theorem in Non-Archimedean Geometry

A version of the fundamental Calabi-Yau theorem in complex analytic geometry states that if $L$ is an ample line bundle on a smooth, complex projective variety $X$, then any smooth volume form on $X$ of the same total mass as $L$ is the curvature volume form of a unique smooth metric on $L$. Recently Yuan and Zhang proved a version of the uniqueness statement in non-Archimedean geometry. I will report on joint work with S. Boucksom and C. Favre where we prove the corresponding existence result. Our method requires us to use a notion of singular semipositive metrics in a non-Archimedean context.

##### An Extension of Hardy-Littlewood-Sobolev Inequality

The classical Hardy-Littlewood-Sobolev inequality implies Sobolev and other geometric inequalities and has a great impact to geometric analysis. Recent results on Sobolev inequalities with negative power indicate the existence of certain extensions to the classical HLS inequality. In this talk, we shall reveal one extension. The existence of sharp constant, as well as the classification of certain extremal functions via the method of moving sphere are also obtained. This is a joint work with J. Dou.

##### Mean curvature flow of two-convex hypersurfaces

We describe recent progress on the mean curvature evolution of smooth, closed, two-convex hypersurfaces in Euclidean space. In this setting, there are two distinct global interpretations of the flow, namely the Huisken-Sinestrari surgery program and the well-known "weak" evolution. We explain the relationship between these two solutions, and present applications of our construction to regularity theory for mean curvature flow.

##### On some nonlocal elliptic and parabolic equations

We show some results on existence and compactness of solutions of a fractional Nirenberg problem. Regularity properties for solutions of some degenerate elliptic equations as well as a Liouville type theorem are established, and used in our blow up analysis. We also introduce a fractional Yamabe flow and show that on the conformal spheres $(S^n, [g_{S^n}])$ it converges to the standard sphere up to a Mobius diffeomorphism. These arguments can be applied to obtain extinction profiles of solutions of some fractional porous medium equations, which are further used to improve a Sobolev inequality via a quantitative estimate of the remainder term.

##### Complex Monge-Ampere type fully nonlinear equations on Hermitian manifolds

In this talk I report some recent results on a class of fully nonlinear elliptic equations on Hermitian manifolds. We shall focus on showing how subsolutions play an important role in deriving apriori estimates. The gradient estimates are new even in the Kahler case.