Yale Geometric Analysis Seminar

Abstract: We discuss the strategies and results on the determination of index and nullity of minimal surfaces. In particular, we prove that for any large enough $m$, the genus $\gamma=m+1$ equator-poles minimal surface doubling of the equatorial two-sphere in the round three-sphere, which has two catenoidal bridges at the poles and  $m$ bridges equidistributed along the equatorial circle and was discovered in earlier work of Kapouleas, has index $2g+5=2m+7$ and nullity $6$.  We also discuss the progress in the study of indices of minimal surfaces from similar constructions.

Abstract:  There is by now a broad body of work on minimal surfaces in positively curved ambient manifolds. If the ambient manifold has nonpositive curvature, much less is known. I will present some recent results on minimal submanifolds in nonpositively curved locally symmetric spaces, that are motivated by or have parallels to the positive curvature setting. The proofs bring new tools into the picture from representation theory. Another key ingredient is a new monotonicity formula for minimal submanifolds of low codimension in nonpositively curved symmetric spaces. 

I will then discuss applications to a program initiated by Gromov to prove statements of the following kind: Suppose we are given two manifolds X and Y, where X is “complicated” and Y is lower dimensional. Then any map f: X-> Y must have at least one “complicated” fiber. If time permits, I will also discuss some applications to systolic geometry, global fixed point statements for actions of lattices on contractible CAT(0) simplicial complexes, and/or non-abelian higher expansion and branched cover stability.  

Abstract: In this talk, I will discuss a series of works on Gromov's p-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index p. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the p-widths and how surfaces provide a convenient setting to investigate this.  

Abstract: Mean curvature flow, the gradient flow of the area functional, is the most natural geometric heat flow for embedded hypersurfaces. Being non linear, the flow develops singularities, at which it stops being smooth. One fundamental, often delicate, question for such non linear flows is that of backwards uniqueness. In this talk I will discuss recent backwards uniqueness results, obtained jointly with Josh Daniels-Holgate, which can address some singularities. I will also compare these results to (commonly more robust) forward uniqueness results, and also to the situation in  other equations.    

Abstract: In this talk, I will discuss the geometry of smooth complete manifolds where the first eigenvalue of the operator −γΔ + Ric, for γ > 0, is bounded from below. Here, Ric denotes the (pointwise) lowest eigenvalue of the Ricci tensor. This condition is weaker than a pointwise lower bound on Ricci curvature. In particular, I will focus on the following result. Let (Mⁿ,g) be a complete non-compact Riemannian manifold with at least two ends, and with n≥2. Assume u is a positive function on M satisfying −γΔu+Ric*u≥0 and γ<4/(n-1). Then, Mⁿ is isometric to the product of IR x Nⁿ⁻¹. This extends a result of Cheeger--Gromoll from 1971, and the bound required on γ is sharp. Joint work with M. Pozzetta and K. Xu.

Abstract: Understanding singularity formation is an important topic in the study of geometric flows. Since Gage-Hamilton-Grayson’s foundational results, it has largely been unknown how singularities of curve shortening flow form in higher codimensions. In this talk, I will present my recent results that in n dim Euclidean space, any curve with a one-to-one convex projection onto some 2-plane develops a Type I singularity and becomes asymptotically circular under curve shortening flow. As a corollary, an analog of Huisken's conjecture for curve shortening flow is confirmed, in the sense that any closed immersed curve in n dim Euclidean space can be perturbed in n+2 dim Euclidean space to a closed immersed curve which shrinks to a round point under curve shortening flow. 

Abstract: In this talk, we will study the mean curvature flow in R^3.  We show that the intrinsic diameter of the surface along mean curvature flow is uniformly bounded as one approaches the first singular time, which confirms a conjecture of Haslhofer. In addition, we establish several sharp quantitative estimates of mean curvature flow. This is a joint work with Yiqi Huang(MIT).  

The uniqueness of infinity plays a crucial role in understanding solutions to geometric PDEs and the geometric and topological properties of manifolds with Ricci curvature bounds. A major progress is made by Colding—Minicozzi who studied one type of infinity, called the asymptotic cones, of a Ricci flat manifold with Euclidean volume growth and proved the uniqueness of asymptotic cones if one cross section is smooth. Their result generalizes an earlier uniqueness result by Cheeger—Tian which requires integrability of the cross section among other things. 

In this talk I will talk about some recent results on the study of the similar uniqueness problem of another type of infinity called the asymptotic limit spaces for Ricci flat manifold with linear volume growth, following the path of Cheeger—Tian and Colding—Mincozzi for cones. Here one considers translation limits instead of rescaling limits and the limits are no longer cones but cylinders. I will highlight the similarities and differences between the two settings. This is joint work with Zetian Yan.

Starting from the celebrated results of Eells and Sampson, a rich and flourishing literature has developed around equivariant harmonic maps from the universal cover of Riemann surfaces into nonpositively curved target spaces. In particular, such maps are known to be rigid, in the sense that they are unique up to natural equivalence. Unfortunately, this rigidity property fails when the target space has positive curvature, and comparatively little is known in this framework. In this talk, given a closed Riemann surface with strictly negative Euler characteristic and a unitary representation of its fundamental group on a separable complex Hilbert space H which is weakly equivalent to the regular representation, we aim to discuss a lower bound on the Dirichlet energy of equivariant harmonic maps from the universal cover of the surface into the unit sphere S of H, and to give a complete classification of the cases in which the equality is achieved. As a remarkable corollary, we obtain a lower bound on the area of equivariant minimal surfaces in S, and we determine all the representations for which there exists an equivariant, area-minimizing minimal surface in S. The subject matter of this talk is a joint work with Antoine Song (Caltech) and Xingzhe Li (Cornell University).

The uniqueness of tangent flows is central to understanding singularity formation in geometric flows. A foundational result of Colding and Minicozzi establishes this uniqueness at cylindrical singularities under the Type I assumption in the Ricci flow. In this talk, I will present a strong uniqueness result for cylindrical tangent flows at the first singular time. Our proof hinges on a Łojasiewicz inequality for the pointed $\mathcal{W}$-entropy, which is established under the assumption that the local geometry near the base point is close to a standard cylinder or its quotient. This is joint work with Yu Li. 

We formulate a stable Bernstein problem in two types of positively curved manifolds. The problem is completely solved for one type in all dimensions and partially solved for the other in low dimensions, leading to several interesting corollaries. 

 In this talk, we will talk about the proof of that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. As a corollary, it implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. This addresses a question posed by Evans-Spruck and De Giorgi. The talk is based on the joint work with Tang-Kai Lee. 

Quantitative unique continuation states that eigenfunctions of the Laplacian on smooth closed manifolds cannot vanish faster than exponentially in its eigenvalue in any open set. It is interpreted physically that the probability of a quantum particle to appear in the classically forbidden region (total energy < potential) is at least exponentially small, also known as quantum tunnelling. In this informal talk, we will discuss the strategy to prove it on both closed and open manifolds with specified end structures (like cones or cylinders) and discuss open problems. 

Abstract: This talk will discuss regularity results for harmonic maps into a certain kind of metric space, called Euclidean buildings.  This will include a discussion of what these spaces are, the behavior of harmonic maps into these spaces, and the connection to superrigidity results for lattices.  We will discuss previously known regularity results before presenting a recent theorem of Breiner, Mese, and myself which generalizes these considerably.