# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### Minimal surfaces with bounded index

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal surfaces with bounded index on a given three-manifold might degenerate. We then discuss several applications, including some compactness results. (This is joint work with O. Chodosh and D. Ketover)

##### Krylov-Evans type theorem for twisted Monge-Amp\'ere equations

Motivated by the pluriclosed flow of Streets and Tian, we establish Evans-Krylov type estimates for parabolic "twisted" Monge-Ampere equations in both the real and complex setting. In particular, a bound on the second derivatives on solutions to these equations yields bounds on Holder norms of the second derivatives. These equations are parabolic but neither not convex nor concave, so the celebrated proof of Evans-Krylov does not apply. In the real case, the method exploits a partial Legendre transform to form second derivative quantities which are subsolutions. Despite the lack of a bona fide complex Legendre transform, we show the result holds in the complex case as well, by formally aping the calculation. This is joint work with Jeff Streets.

##### Compactness questions for triholomorphic maps

A triholomorphic map u between hyperKahler manifolds solves the "quaternion del-bar" equation du=I du i + J du j + K du k. Such a map turns out, under suitable assumptions, to be stationary harmonic. We focus on compactness issues regarding the quantization of the Dirichlet energy and the structure of the blow-up set. We can relax the assumptions on the manifolds, in particular we can take the domain to be merely "almost hyper-Hermitian": this more general setting leads to the weaker notion of"almost-stationarity", without however affecting our compactness results and it leads e.g. to gauge-theoretic applications. This is a joint work with G. Tian (Princeton).

##### Sharp Trace-Sobolev inequalities of order 4

We establish sharp Sobolev inequalities of order four on Euclidean d-balls for d greater than or equal to four. When d=4, our inequality generalizes the classical second order Lebedev-Milin inequality on Euclidean 2-balls. Our method relies on the use of scattering theory on hyperbolic d-balls. As an application, we characterize the extremals of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean 4-balls. This is joint work with Alice Chang.

##### Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic maps - Part 1

This series of talks is an extension of the colloquium talk where we outline a proof of recent results concerning the structure of stationary and minimizing harmonic maps. Specifically, we will show that the singular stratum S^k(f) are k-rectifiable for a stationary map, and for a minimizing map that the singular set S(f) has finite n-3 measure. We will also show sharp sobolev estimates for solutions.

##### Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic maps (Part 2)

This series of talks is an extension of the colloquium talk where we outline a proof of recent results concerning the structure of stationary and minimizing harmonic maps. Specifically, we will show that the singular stratum S^k(f) are k-rectifiable for a stationary map, and for a minimizing map that the singular set S(f) has finite n-3 measure. We will also show sharp sobolev estimates for solutions.

##### A proportionality of scalar curvatures on Hermitian manifolds and Schrödinger operators

**Please note special day, time and location.** On a Kähler manifold there is a clear connection between the complex geometry and underlying Riemannian geometry. However, in the non-Kähler setting, such a link is not so obvious. I will discuss the existence of non-Kähler Hermitian metrics for which a certain proportionality relationship between the Chern and Riemannian scalar curvatures holds. The study of such metrics, in turn, leads to a general question concerning the behavior of the lowest eigenvalue of Schrödinger operators on compact Riemannian manifolds. This is joint work with Mike Dabkowski.

##### Motion by curvature in the subriemannian Heisenberg group

In this talk we will present some properties of the motion by curvature in subriemannian setting. This describes the motion of a surface when each point is moving in the normal direction with speed proportional to the mean curvature. The problem is well undestood in Riemannian setting, while only partial results are known in the subriemannian one, due to the presence of characteristic points of the evolving surface. In order to avoid the problem solutions can be found as limit of the riemannian approximation. We present here some existence and uniqueness results for these type of solutions, obtained in collaboration with E. Baspinar, and some application to image completion obtained in collaboration with Franceschello, Sanguinetti and Sarti.

##### Minimal surfaces of finite total curvature in Hˆ2xR

**Please note special time (4:15). ** In this talk we will present some recent developments in the theory of finite total curvature minimal surfaces in Hˆ2xR and we will show a characterization for such surfaces in terms of their behavior at infinity. This is a joint work with Hauswirth and Rodriguez.

##### An index theorem for CR manifolds with S^{1} action

**Please note special day, location and time. **

##### Bounds on eigenvalues on riemannian manifolds

##### Ricci solitons

**Please note special time (4:15). **We discuss the asymptotic geometry of complete noncompact four dimensional shrinking Ricci solitons and outline a program for their classification.

##### Positivity of the complex Neumann Laplacian

In this talk we will discuss aspects of spectral theory of the complex Neumann Laplacian in several complex variables. In particular, we will discuss geometric and potential theoretic characterizations of positivity and spectral discreteness of the complex Neumann Laplacian.

##### Min-max, phase transitions and minimal hypersurfaces

**Please note special time. **There is a strong correspondence between critical points of in the theory of phase transitions and critical points of the area functional in theory of minimal hypersurfaces. Historically, research has focused in studying the case of minima or stable critical points. We use ideas from Pitts to extend these results to the case of unstable critical points of any index. As an application we obtain a new min-max method for constructing embedded minimal hypersurfaces in an arbitrary closed manifold of any dimension. Our approach is variational, but it is substantially different from Almgren-Pitts theory. We also study the correspondence of critical points from a "global" variational point of view in the case of those obtained by min-max methods.

##### Regularity theory for fully nonlinear integro-differential equations

**Please note special day (Thursday) and room (Fine1001). **Integro-differential equations appear naturally when studying discontinuous stochastic processes, and we are interested in the regularity properties of their solutions. In this talk we will prove Schauder estimates for fully nonlinear integro-differential equations, where a key ingredient is a recursive Evans-Krylov theorem for translation invariant equations. This is joint work with Jingang Xiong.** **

##### A Riemannian structure on the space of conformal metrics

**Please note special day (Thursday) and room (Fine 601). ** I will describe a Riemannanian structure on the space of conformal metrics satisfying a certain positivity condition. This metric is inspired by the Riemannian of the space of Kahler metrics, and shares many of the same properties. After defining the metric and deriving the geodesic equation, I will specialize to the case of two dimensions and prove some of the basic properties of the space. This is joint work with J. Streets (UC-Irvine).

##### On the tangent cones in collapsed limit spaces with lower Ricci curvature bound

We will discuss open questions and some recent results regarding the tangent cones in collapsed limit spaces of manifolds with lower Ricci curvature bounds.

##### Asymptotic shapes of neckpinch singularities in geometric flows

Geometric flows such as Ricci flow and mean curvature flow are nonlinear parabolic PDEs that tend to develop singularities in finite time. A useful approach to analyzing singularities is the technique of matched asymptotics, which can provide detailed and precise information including the rate of curvature blow-up, the set of points where singularity forms, and the behavior of the solution in a space-time neighborhood of that singularity. In this talk, we will survey the results concerning the asymptotic shapes of neckpinch singularities in Ricci flow and mean curvature flow.

##### Tian's properness conjectures, the strong Moser-Trudinger inequality, and infinite-dimensional Finsler geometry

In the 90's, Tian introduced a notion of properness in the space of Kahler metrics in terms of Aubin's nonlinear Dirichlet energy and Mabuchi's K-energy and put forward several conjectures on the relation between properness and existence of Kähler-Einstein metrics. These can be viewed as the Kahler analog of the classical Moser-Trudinger inequality from conformal geometry. In joint work with Y. Rubinstein we disprove one of

these conjectures, and prove the remaining ones. Our techniques are flexible and extend to many different situations, including Kahler-Einstein edge metrics and Kahler-Ricci solitons. Moreover, we formulate a

corresponding conjecture for constant scalar curvature metrics and reduce it to a PDE regularity problem of certain weak minimizers of the K-energy.

##### A nonlocal diffusion problem on manifolds

**Please note special day, time and location. ** We consider a nonlocal diffusion problem on a manifold. This kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. After briefly considering existence and uniqueness of solutions, we prove that, for a convenien rescaling the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior: while on compact manifolds the asymptotics are given by the spectral properties of the operator; in the model of hyperbolic space we find a different and interesting behavior. This is joint work with C. Bandle, M. Fontelos and N. Wolanski.