Registration

Chung-Jun Tsai 蔡忠潤

Title: Bernstein theorems for calibrated submanifolds in ℝ7 and ℝ8

Abstract: The classical Bernstein theorem asserts that a complete minimal surface which is the graph of f: ℝ2 → ℝ1 must be a plane. There have been many studies on this Bernstein type theorem in the general setting.

In this talk, I will explain a Bernstein theorem for f: ℝ4 → ℝ3 whose graphs form coassociative submanifolds in ℝ7, and also for Cayley submanifolds in ℝ8. This is based on a joint work with Chun-Kai Lien.

Lunch break

Pak-Yeung Chan 陳柏揚

Title: A family of Kahler flying wing steady Ricci solitons

Abstract: Ricci solitons are generalization of the Einstein manifolds and are self similar solutions of the Ricci flow. In particular, steady Kahler-Ricci solitons are eternal solutions of the Kahler-Ricci flow. In this talk, we will discuss some recent examples of such solitons with positive sectional curvature on C^n, where n>2. This is based on a joint work with Ronan Conlon and Yi Lai.

Tea break

Wei-Yi Chiu 邱維毅

Title: Volumetric Minkowski inequality on manifolds with weighted Poincaré inequality

Abstract: In this talk, I will motivate the study of volumetric Minkowski inequalities in the context of Riemannian manifolds equipped with a weighted Poincaré inequality. These inequalities naturally arise in various geometric settings where the Ricci curvature is bounded below by a function rather than a constant. I will begin by reviewing classical results in Euclidean and curved spaces, and explain how the notion of weighted Poincaré inequality offers a unified framework to extend such results.

Then, I will present our main theorem: an upper bound for the weighted volume of compact smooth domains in terms of the boundary mean curvature. While the full proof involves a transformation to a weighted metric measure space and the use of a weighted Laplacian, in this presentation I will only sketch the key steps by applying and recalling some essential lemmas, to highlight the core ideas without going into technical details.

Nobuhiro Morita 森田展弘

Title: Heat Kernel Asymptotics for Degenerate Weight Functions on C^n

Abstract: The study of the heat kernel asymptotics for high power of line bundles is closely related to some important problems in complex analysis and geometry. In this talk, we will first explain why heat kernel is important and introduce Bismut's heat kernel asymptotics. Next, we generalize Bismut's results to degenerate weight functions on C^n.

Forum discussion

Symposium Banquet

Place: 國立清華大學 風雲樓 (within the campus)