# UCSB Differential Geometry Seminar

The talks are held on every Friday from 3:00 to 4:00 PM in South Hall 6635 (unless otherwise noted).

## Spring 2020

Friday, April 10: Junrong Yan on Zoom
Title : Witten deformation on noncompact manifolds and Mirror Symmetry for Landau-Ginzburg model

Abstract: In 1987, Witten introduced a deformation of the de Rham complex by considering the new differential $d+df$, where $f$ is a (Morse) function. Since then, Witten deformation has found important applications such as Bismut-Zhang/Cheeger-Muller theorem as well as being instrumental in the development of Floer homology theory. To understand the mathematics of Landau-Ginzburg model, we study the Witten deformation for noncompact manifolds. We will begin with some brief reviews on the physical origin of Witten deformation, Morse theory and then move on to the noncompact case. We will discuss the $L^2$ cohomology of $d+df$, the index theorem and analytic torsion of Witten Laplacian as well as their relations with the Mirror symmetry for Landau-Ginzburg model.

Friday, May 1: Eric Chen on Zoom
Title: $L^{n/2}$ curvature pinching of Yamabe metrics

Abstract: I will discuss how the control of Sobolev constants under a Ricci flow starting from Yamabe metrics suitably pinched in an integral norm sense allows us to prove such manifolds must be diffeomorphic to space forms, generalizing work of Gursky and Hebey–Vaugon. This is joint work with Guofang Wei and Rugang Ye.

Friday, May 8: Xin Zhou on Zoom
Title: Generic scarring for minimal hypersurfaces

Abstract: In classical spectral theory, Equidistribution and Scarring, which are two significant but opposite phenomena, concern the distribution of normalized energy measures for Laplacian eigenfunctions on closed manifolds. The Quantum Ergodicity asserts that in negative curvature almost all subsequence of Laplacian eigenfunctions have their normalized energy measures equidistributing, while Scarring means that some particular subsequence of normalized energy measures concentrate on proper subsets. In this talk, we will present a scarring phenomenon for minimal hypersurfaces for a generic set of smooth metrics. This is a joint work with Antoine Song.

Friday, June 5: Rick Ye on Zoom
Title: Ricci flow, $L^{n/2}$-Sobolev Almost Flat Manifolds and Gromov Almost Flat Manifolds

Abstract: The celebrated Gromov Almost Flat Manifold Theorem states that a compact manifold of dimension $n$ is diffeomorphic to an infranil manifold if and only if it admits an almost flat metric, i.e. a Riemannian metric satisfying $|\mathrm{Rm}|\mathrm{diam}^2\le\epsilon(n)$, where $\mathrm{Rm}$ denotes the Riemann curvature tens or, $\mathrm{diam}$ the diameter, and $\epsilon(n)$ a dimensional positive constant. In this joint work with Eric Chen and Guofang Wei we prove that a compact manifold of dimension $n$ is diffeomorphic to an infranil manifold if and only if it admits an $L^{n/2}$-Sobolev almost flat metric, i.e. a Riemmannian metric satisfying $|\mathrm{Rm}|_{n/2}C_S^3\le\epsilon(n)$, where $|\mathrm{Rm}|_{n/2}$ is the $L^{n/2}$ norm of $\mathrm{Rm}$, and $C_S$ the Sobolev constant.

## Winter 2020

Friday, January 31: Darong (Daren) Cheng
Title: Bubble tree convergence of conformally cross product preserving maps

Abstract: We study a class of weakly conformal $3$-harmonic maps, called Smith maps, which parametrize associative $3$-folds in $7$-manifolds equipped with $G_2$-structures. These maps satisfy a first-order system of PDEs generalizing the Cauchy-Riemann equation for $J$-holomorphic curves, and we are interested in their bubbling phenomena. Specifically, we first prove an $\epsilon$-regularity theorem for Smith maps in $W^{1, 3}$, and then explain how that combines with conformal invariance to yield bubble trees of Smith maps from sequences of such maps with uniformly bounded $3$-energy. When the $G_2$-structure is closed, we show that both $3$-energy and homotopy are preserved in the bubble tree limit. The result can be viewed as an associative analogue of the bubble tree convergence theorem for $J$-holomorphic curves. This is joint work with Spiro Karigiannis and Jesse Madnick.

Friday, Feburary 21: Jiayin Pan
Title: On the escape rate of geodesic loops in an open manifold with nonnegative Ricci curvature

Abstract: A consequence of the Cheeger-Gromoll splitting theorem states that for any open manifold $(M,x)$ of nonnegative Ricci curvature, if all the minimal geodesic loops at $x$ that represent elements of $\pi_1(M,x)$ are contained in a bounded ball, then $\pi_1(M,x)$ is virtually abelian. We generalize the above result: if these minimal representing geodesic loops escape from any bounded metric balls at a sublinear rate with respect to their lengths, then $\pi_1(M,x)$ is virtually abelian.

Monday, Feburary 24, 2:00 to 3:00 PM: Yongbin Ruan
Title: BCOV axioms of Gromov-Witten theory of Calabi-Yau $3$-fold

Abstract: One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau $3$-fold such as the quintic $3$-folds. There have been a collection of remarkable conjectures from physics (BCOV B-model) regarding the universal structure or axioms of higher genus Gromov-Witten theory of Calabi-Yau $3$-folds. In the talk, I will first explain 4 BCOV axioms explicitly for the quintic $3$-folds. Then, I will outline a solution for 3+1/2 of them. This talk is based on the joint works with Q. Chen, F. Janda and S. Guo.

Friday, March 6: Yangyang Li
Title: Existence of minimal hypersurfaces with arbitrarily large area and Morse index

Abstract: The recent development of Almgren-Pitts min-max theory has presented the abundance of minimal hypersurfaces. In particular, in a bumpy closed Riemannian manifold $(M^{n+1}, g)$ $(3\leq n+1\leq 7)$, X. Zhou’s multiplicity one theorem and Marques-Neves Morse index theorem lead to the fact that for each positive integer $p$, there exists a minimal hypersurface with Morse index $p$ and area propotional to $p^{1/(n+1)}$. However, the minimal hypersurface here might have multiple disjoint connected components, so the geometric complexity of it could merely be the accumulation of its components with relatively small area or Morse index, and it is not clear whether at least one component would have such geometric complexity. In this talk, by adapting the theorems mentioned above into a confined min-max setting, we will show that such a manifold always admits a sequence of connected closed embedded two-sided minimal hypersurfaces whose areas and Morse indices both tend to infinity. This improves a previous result by O. Chodosh and C. Mantoulidis on connected minimal hypersurfaces with arbitrarily large area.

## Fall 2019

Friday, October 4: Zhichao Wang
Title: Compactness of self-shrinkers in $\mathbb R^3$ with fixed genus

Abstract: In this talk, we present the compactness of self-shrinkers in $\mathbb R^3$ with bounded entropy and fixed genus. As a corollary, we show that numbers of ends of such surfaces are uniformly bounded by the entropy and genus. This is a joint work with Ao Sun.

Friday, October 11: Xin Zhou
Title: Multiplicity One Conjecture in Min-max theory

Abstract: I will present the proof of Multiplicity One Conjecture in Min-max theory raised by Marques and Neves. It says that in a closed manifold of dimension between $3$ and $7$ with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one.

Friday, October 18: Eric Chen
Title: Integral pinching for the Ricci flow on asymptotically flat manifolds

Abstract: Many pinching results for the Ricci flow, which guarantee its long-time existence and convergence under initial curvature restrictions, have been proved in the compact case, including Hamilton’s foundational results and Brendle-Schoen’s differentiable sphere theorem. Less is known in the noncompact case. I will discuss the proof of a pinching result for the Ricci flow on asymptotically flat manifolds based on an initial integral curvature restriction.

Friday, October 25: Christian Rose
Title: Recent developments for Kato type Ricci curvature conditions

Abstarct: The Kato condition is a tool from perturbation theory of Dirichlet forms to control perturbed heat semigroups. Using this as a more general condition than $L^p$-bounds for the negative part of Ricci curvature, I will discuss several recent results in part obtained with Gilles Carron from Nantes, such as Lichnerowicz, Cheeger, and isoperimetric constant estimates for compact manifolds as well as a very recent generalization of Myers’ compactness theorem.

Tuesday, November 5, 4-5 pm at SH63617: Ao Sun
Title: Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

Abstract: One of the central topics in mean curvature flow is understanding the singularities. In 1995, Ilmanen conjectured that the first singularity appeared in a smooth mean curvature flow of surfaces must have multiplicity one. Following the theory of generic mean curvature flow developed by Colding-Minicozzi, we prove that a closed singularity with high multiplicity is not generic, in the sense that we may perturb the flow so that this singularity with high multiplicity can never show up. One of the main techniques is the local entropy, which is an extension of the entropy used by Colding-Minicozzi to study the generic mean curvature flow.

Friday, November 15: Fedor Manin
Title: Scalable spaces

Abstract: Given a Riemannian manifold $M$, harmonic forms induce a map $H^*(M;\mathbb{R}) \to \Omega^*(M)$ which is in general not multiplicative. Manifolds for which it is are called geometrically formal and besides compact symmetric spaces few examples are known. A different picture emerges when we ask about the existence of some multiplicative map $H^*(M;\mathbb{R}) \to \Omega^*(M)$. For example, such a map exists for $(\mathbb{CP}^2)^{\#3}$ but not $(\mathbb{CP}^2)^{\#4}$. Unlike geometric formality, this property is purely topological, in fact an invariant of rational homotopy type. It also has a number of geometric and topological consequences and equivalent formulations; in particular such spaces are special from the point of view of Lipschitz homotopy theory. This is joint work with Sasha Berdnikov.

Friday, December 6: Zhifei Zhu
Title: Geometric Inequalities on Riemannian manifolds with bounded Ricci curvature

Abstract: I will discuss some upper bounds for the length of a shortest periodic geodesic, and the smallest area of a closed minimal surface on closed Riemannian manifolds of dimension $4$ with Ricci curvature between $-1$ and $1$. These are the first bounds that use information about the Ricci curvature rather than sectional curvature of the manifold. (Joint with Nan Wu).