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Hung-Lin Chiu 邱鴻麟

Title: Constant p-mean curvature hypersurfaces in the Heisenberg groups H_n

Abstract: Alexandrov's soap bubble theorem asserts that spheres are the only connected closed embedded hypersurfaces in the Euclidean space with constant mean curvature. Alexandrov's type theorem is also a fundamental problem, and, as far as I know, is still open in the Heisenberg groups of higher CR dimensional cases. In this talk we introduce our study on the constant p-mean curvature hypersurfaces in the Heisenberg groups. If n=1, we will show that the existence of a constant p-mean curvature surface is equivalent to the existence of a solution to a nonlinear second-order ODE. In some sense, they are in one-to-one correspondence. As a result, after a kind of normalization, we obtain a representation of constant p-mean curvature surfaces and classify further all constant p-mean curvature surfaces. For higher CR dimensional cases, the situation is totally different. We thus focus on the umbilic hypersurfaces in the Heisenberg groups. We show the fundamental theorems for rotationally symmetric hypersurfaces, and thus, together with our earlier results, provide a complete classification of umbilic hypersurfaces in the Heisenberg groups. In addition, we give a complete description of generating curves for rotationally symmetric hypersurfaces with constant p-mean curvature in the Heisenberg groups. In particular, we thus establish the validity of Alexandrov's type theorem for umbilic hypersurfaces in the Heisenberg groups. This reduces the Alexandrov's type conjecture to be whether or not it is umbilic.   

Lunch break

Wei-Bo Su 蘇瑋栢, National Central University

Title: Type II finite-time singularities in Lagrangian mean curvature flow

Abstract: A solution to the mean curvature flow undergoes a Type~II finite-time singularity when the singular behavior occurs at scales much smaller than the parabolic scale. However, detecting the exact regularity scale is quite challenging, and therefore it is difficult to determine the blow-up rate of the curvature. In this talk, I will describe recent progress, joint with Maxwell~Stolarski, on a constructive approach to this problem. In particular, for each integer $K \ge 2$, we construct a solution to the Lagrangian mean curvature flow that develops a Type~II finite-time singularity at time $T$, where the curvature blows up like $(T - t)^{-K/2}$ as $t \to T$. The novelty of our construction is that, instead of perturbing the tangent flow by higher modes and then matching it with a Type~II model, we directly excite the ``ground state'' (namely, the special Lagrangian Type~II model) to higher modes. This overcomes the difficulty that the special Lagrangian Type~II model cannot be matched with any $L^{2}$ eigenfunctions on its asymptotic cone.

Tea break

Kui-Yo Chen 陳奎佑

Title: On the Coordinate Ring of SL(3,ℂ)-Character Varieties under Poisson Reduction

Abstract: This talk is about the character varieties associated with the group SL(3,ℂ), focusing on its maximal compact subgroup SU(3). Specifically, we examine the structure and properties of these varieties over a once-punctured torus.

Ming Hsiao 蕭明

Title: Curvature estimates for steady and expanding solitons in higher dimensions

Abstract: As natural generalizations of Einstein manifolds and self-similar solutions to the Ricci flow, Ricci solitons have been extensively studied over the past decades. A central question is to understand the curvature behavior under mild geometric assumptions. In this talk, I will present two new curvature estimates for steady and expanding solitons, respectively. This is a joint work with Pak-Yeung Chan.

Forum discussion

Symposium Banquet

Place: 73階蔬食咖啡