The radiation field on product cones

Schauder estimates on products of cones

We prove an interior Schauder estimate for the Laplacian on metric products of two dimensional cones with a Euclidean factor, generalizing the work of Donaldson and reproving the Schauder estimate of Guo-Song. We characterize the space of homogeneous subquadratic harmonic functions on products of cones, and identify scales at which geodesic balls can be well approximated by balls centered at the apex of an appropriate model cone. We then locally approximate solutions by subquadratic harmonic functions at these scales to measure the Hölder continuity of second derivatives.

Stability of sin-cones and cosh-cylinders

This work concerns stability and instability of Einstein warped products with an Einsteinian fiber of codimension 1. We study the cases where the scalar curvature of the warped product and of the fiber are either both positive or both negative to complement the results in [Krö16]. Up to a small gap in the case of sin-cones, the stability properties of such warped products are now completely determined by spectral properties of the Laplacian and the Einstein operator of the fiber. For cosh-cylinders, we are furthermore able to prove a convergence result for the Ricci flow starting in a small ne…

Riemann 流形凸锥中的 Serrin 型问题

Serrin's type Problems in convex cones in Riemannian manifolds

本文研究 Riemann 流形凸锥中的 Serrin 型超定问题，建立扭曲乘积与 Einstein 情形的刚性结果、肥皂泡定理及 Heintze–Karcher 不等式，并讨论 Euclid 锥上漂移 Laplace 算子的类似问题。

锥体和扭曲积流形上的热点

Hot spots on cones and warped product manifolds

研究了扭曲积流形上热方程热点的极值问题及长期行为，证明了Rauch猜想在特定条件下成立，并揭示了谱间隙对长期行为的影响。

乘积锥上薛定谔方程的色散估计与最优性

Dispersive estimates and optimality for Schrödinger equations on product cones

本文研究乘积锥上薛定谔传播子的时间衰减估计，证明当半径$\rho\geq1$时通常色散估计成立，否则不成立，方法基于调和分析，补充并优化了Jia-Zhang的微局部分析结果。

Resolvent and spectral measure for Schrödinger operators on flat Euclidean cones

We construct the Schwartz kernel of resolvent and spectral measure for Schrödinger operators on the flat Euclidean cone $(X,g)$, where $X=C(\mathbb{S}_σ^1)=(0,\infty)\times \mathbb{S}_σ^1$ is a product cone over the circle, $\mathbb{S}_σ^1=\R/2πσ\Z$, with radius $σ>0$ and the metric $g=dr^2+r^2 dθ^2$. As products, we prove the dispersive estimates for the Schrödinger and half-wave propagators in this setting.