Strictly pseudoconvex domains in C^2 with obstruction flat boundary

Speaker: 

Sean Curry

Institution: 

UC San Diego

Time: 

Tuesday, October 30, 2018 - 4:00pm

Location: 

RH 306

A bounded strictly pseudoconvex domain in C^n, n>1, supports a
unique complete Kahler-Einstein metric determined by the Cheng-Yau solution
of Fefferman's Monge-Ampere equation. The smoothness of the solution of
Fefferman's equation up to the boundary is obstructed by a local CR
invariant of the boundary called the obstruction density. In the case n=2
the obstruction density is especially important, in particular in describing
the logarithmic singularity of the Bergman kernel. For domains in C^2
diffeomorphic to the ball, we motivate and consider the problem of
determining whether the global vanishing of this obstruction implies
biholomorphic equivalence to the unit ball. (This is a strong form of the
Ramadanov Conjecture.)

Convex hypersurface theory in higher-dimensional contact topology

Speaker: 

Ko Honda

Institution: 

UCLA

Time: 

Tuesday, November 20, 2018 - 4:00pm

Location: 

RH 306

Convex surface theory and bypasses are extremely powerful tools
for analyzing contact 3-manifolds.  In particular they have been
successfully applied to many classification problems.  After reviewing
convex surface theory in dimension three,  we explain how to generalize many
of their properties to higher dimensions.   This is joint work with Yang
Huang.

The Minkowski formula and the quasi-local mass

Speaker: 

Po-Ning Chen

Institution: 

UC Riverside

Time: 

Tuesday, November 6, 2018 - 4:00pm to 5:00pm

Location: 

RH 306

In this talk, we will discuss the relationship between the Minkowski formula and the quasi-local mass in general relativity, In particular, we will use the Minkowski formula to estimate the quasi-local mass. Combining the estimate and the positive mass theorem, we obtain rigidity theorems which characterize the Euclidean space and the hyperbolic space.

The degenerate special Lagrangian equation on Riemannian manifolds.

Speaker: 

Matthew Dellatorre

Institution: 

University of Maryland

Time: 

Tuesday, November 13, 2018 - 4:00pm

Location: 

RH 306

We show that the degenerate special Lagrangian equation (DSL), recently introduced by Rubinstein–Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi–Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey–Lawson, in the Riemannian setting to obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and hence continuous geodesics in the space of positive Lagrangians

Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory

Speaker: 

Yi Wang

Institution: 

Johns Hopkins University

Time: 

Tuesday, October 9, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

306 Rowland Hall

We consider $\sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when $2k<n+1$. We explicitly give examples of product manifolds with multiple solutions. It is analogous to Schoen’s example for Yamabe problem on $S^1\times S^{n-1}$. This is joint work with Jeffrey Case and Ana Claudia Moreira.

Bershadsky--Cecotti--Ooguri--Vafa torsion in Landau--Ginzburg models

Speaker: 

Guangbo Xu

Institution: 

SUNY Stony Brook

Time: 

Tuesday, October 9, 2018 - 3:00pm to 4:00pm

Host: 

Location: 

306 Rowland Hall

In the celebrated work of Bershadsky--Cecotti--Ooguri--Vafa the genus one string partition function in the B-model is identified with certain analytic torsion of the Hodge Laplacian on a K\"ahler manifold. In a joint work with Shu Shen (IMJ-PRG) and Jianqing Yu (USTC) we study the analogous torsion in Landau--Ginzburg models. I will explain the corresponding index theorem based on the asymptotic expansion of the heat kernel of the Schr\"odinger operator. I will also explain the rigorous definition of the BCOV torsion for homogeneous polynomials on ${\mathbb C}^N$. Lastly I will explain the conjecture stating that in the Calabi--Yau case the BCOV torsion solves the holomorphic anomaly equation for marginal deformations.

The Morse index of Free Boundary Minimal Hypersurfaces

Speaker: 

Hung Tran

Institution: 

Texas Tech University

Time: 

Tuesday, May 22, 2018 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

A free boundary minimal hypersurface in the unit Euclidean ball is a critical point of the area functional among all hypersurfaces with boundaries in the unit sphere, the boundary of the ball. While regularity and existence aspects of this subjecct have been extensively investigated, little is known about uniqueness. That motivates the study of the Morse index, which quantitatively measures the number of deformations decreasing the area to second order. Henceforth, A. Fraser and R. Schoen proposed a fundamental conjecture concerning surfaces with low indices. In this talk, we discuss recent developments including a joint work with Ari Stern, Detang Zhou, and Graham Smith.

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