Yuchin Sun 孫有慶 (UC Santa Cruz): Morse index bound of min-max 2 spheres

Finite-dimensional Morse theory was developed to study geodesics. Index of a critical point of a Morse function reflects its topology. A natural extension of finite-dimensional Morse theory will be a Morse theory for harmonic spheres. We prove that given a Riemannian manifold of dimension at least three, with a generic metric and nontrivial homotopy group \pi_3, there exists a collection of finitely many harmonic spheres whose sum of areas realizes the width with Morse index bound one. Moreover, under the assumption of strong convergence the Morse index conjecture is true.

thesis defense of Wei-Bo Su 蘇瑋栢 (NTU): Stability of Minimal Lagrangian Submanifolds and Soliton Solutions for Lagrangian Mean Curvature Flow

Stability provides important information about critical points of some functionals. In this talk, the functionals we consider are $f$-volume functionals defined on the space of Lagrangian submanifolds in a K\"ahler manifold $X$, where $f$ is a function on $X$. The critical points are $f$-minimal Lagrangian submanifolds, which are generalizations of minimal Lagrangian submanifolds and soliton solutions for Lagrangian mean curvature flow (LMCF). We study two different notions of stability with respect to the $f$-volume functional, namely the linear stability and dynamic stability.

The linear stability concerning the positivity of second variation of $f$-volume functional at an $f$-minimal Lagrangian submanifold. We derive a second variation formula for $f$-minimal Lagrangian submanifolds, which is a generalization of the second variation formula by Chen and Oh. Using this we obtain stability criterions for $f$-minimal Lagrangian submanifolds in gradient K\"ahler--Ricci solitons. In particular, we show that expanding and translating solitons for LMCF are $f$-stable.

The dynamic stability on the other hand regarding the existence and convergence of the negative gradient flow of the $f$-volume functional, the generalized LMCF, starting from an initial data nearby a critical point. Since the examples of $f$-minimal Lagrangians we are most interested in are complete noncompact, we first prove a short-time existence for asymptotically conical (AC) LMCF. Then we give some long-time existence and convergence results for equivariant, almost-calibrated, AC LMCF in $\mathbb{C}^{m}$.

Shih-Kai Chiu 邱詩凱 (Notre Dame): A Liouville type theorem for harmonic 1-forms

The famous Cheng-Yau gradient estimate implies that on a complete Riemannian manifold with nonnegative Ricci curvature, any harmonic function that grows sublinearly must be a constant. This is the same as saying the function is closed as a 0-form. We prove an analogous result for harmonic 1-forms. Namely, on a complete Ricci-flat manifold with Euclidean volume growth, any harmonic 1-form with polynomial sublinear growth must be the differential of a harmonic function. We prove this by proving an L^2 version of the "gradient estimate" for harmoinc 1-forms. As a corollary, we show that when the manifold is Ricci-flat Kähler with Euclidean volume growth, then any subquadratic harmonic function function must be pluriharmonic. This generalizes a result of Conlon-Hein.

Yu-Shen Lin 林昱伸 (Boston University): New examples of special Lagrangian submanifolds in log Calabi-Yau manifolds

Special Lagrangians are important examples of minimal submanifolds in Calabi-Yau manifolds introduced by Harvey-Lawson. In this talk, we will provide some new examples of special Lagrangian submanifolds in certain log Calabi-Yau manifolds via the techniques of Lagrangian mean curvature flow. We will also talk about the applications to mirror symmetry, including the existence of special Lagrangian fibration in some log Calabi-Yau surfaces. This is base on a joint work with T. Collins and A. Jacob.

Da Rong Cheng 程大容 (University of Chicago): Bubble tree convergence of conformally cross product preserving maps

Vector cross products (VCPs) are known to be closely related to calibrated geometry. In particular, certain calibrated submanifolds can be viewed as images of VCP preserving maps. In this talk, we will be interested in the class of "conformally VCP preserving maps" (originally introduced by A. Smith under a different name), which may be thought of as weakly conformal parametrizations of associative (Cayley, resp.) objects in a G_2- (Spin(7)-, resp.) manifold. Our main result is a bubble tree convergence theorem for a sequence of conformally VCP preserving maps with uniformly bounded energy. This is joint work with S. Karigiannis (Waterloo) and J. Madnick (McMaster).

Kai-Wei Zhao 趙凱衛 (University of California, Irvine): The Asymptotic Behavior of Solution of Jang Equation near Closed MOTS and Shape of Interior Regularization Limit

In 1981 Schoen and Yau prove spacetime positive mass theorem by reducing the problem to the time-symmetric case which they prove in 1979. One key ingredient is Jang equation. The obstacles of solutions to be entire are marginally outer trapped surfaces (MOTS). Andersson and Metzger show that closed MOTS where solutions of Jang equation blow up are stable in the sense of MOTS. The first result in this talk is that the graph of solution of Jang equation approaches exponentially to the cylinder of such closed MOTS with strong stability assumption. In addition, Schoen and Yau exploit a family of regularization equations to show the existence and other properties of Jang equation, but they focus only on the region outside MOTS. In this talk we will characterize the geometry of interior regularization limit.