Heat Kernel on Homogeneous Bundles over Symmetric Spaces

Kähler-Ricci flow on homogeneous toric bundles

Assume that $X$ is a homogeneous toric bundle of the form $G^{\mathbb{C}}\times_{P,τ} F$ and is Fano, where $G$ is a compact semisimple Lie group with complexification $G^\mathbb{C}$, $P$ a parabolic subgroup of $G^\mathbb{C}$, $τ:P\rightarrow (T^m)^\mathbb{C}$ is a surjective homomorphism from $P$ to the algebraic torus $(T^m)^\mathbb{C}$, and $F$ is a compact toric manifold of complex dimension $m$. In this note we show that the normalized Kähler-Ricci flow on $X$ with a $G\times T^m$-invariant initial Kähler form in $c_1(X)$ converges, modulo the algebraic torus action, to a Kähler-Ricci so…

The Yang-Mills heat flow on the moduli space of framed bundles on a surface

We study the analog of the Yang-Mills heat flow on the moduli space of framed bundles on a cut surface. Existence and convergence of the heat flow give a stratification of Morse type invariant under the action of the loop group. We use the stratification to prove versions of Kahler quantization commutes with reduction and Kirwan surjectivity.

Gradient estimate for the Poisson equation and the non-homogeneous heat equation on compact Riemannian manifolds

In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.

在齐性向量丛上的规定曲率问题

Prescribed mean curvature problems on homogeneous vector bundles

本文研究了在有理齐性簇上的齐性全纯向量丛上弱奇 Hermite-Einstein 结构的存在性，利用 Cartan 的最高权理论，建立了向量丛 ${f{E}}$ 允许拓扑分解 ${f{E}} \cong {f{E}}_{0} \otimes {f{L}}_{0}$ 的显式代数准则，其中 ${f{L}}_{0} \in { m{Pic}}(X)$ 且 $c_{1}({f{E}}_{0}) = 0$。当此条件满足时，规定曲率方程完全解耦。通过将拓扑障碍完全转移到线丛 ${f{L}}_{0}$ 上，这种分解将非阿贝尔的规定曲率问题减少到 Demailly 的阿贝尔理论中奇异线丛度量的范畴。作为主要应用，我们得到一个以交数表示的充分代数条件，使得一个 $L^{2}$-函数可以被实现为不可约齐性向量丛上奇异 Hermitian 结构的曲率。最终，通过克服经典 Bando-Siu 框架中固有的曲率限制，这种方法提供了一种稳健的机制来构造能容纳沿解析子簇上规定奇点的奇异 Hermitian 结构。

Dirichlet sub-Laplacians on homogeneous Carnot groups: spectral properties, asymptotics, and heat content

We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum, and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.

Refined asymptotics for the infinite heat equation with homogeneous Dirichlet boundary conditions

The nonnegative viscosity solutions to the infinite heat equation with homogeneous Dirichlet boundary conditions are shown to converge as time increases to infinity to a uniquely determined limit after a suitable time rescaling. The proof relies on the half-relaxed limits technique as well as interior positivity estimates and boundary estimates. The expansion of the support is also studied.

Homogeneous algebras via heat kernel estimates

We study homogeneous Besov and Triebel--Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with $L^\infty$ is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings.

Morrey空间上带齐次权重的热半群光滑化估计

Smoothing estimate for the heat semigroup with a homogeneous weight on Morrey spaces

本文在Morrey空间上研究了与Hardy-Hénon抛物方程非线性项相关的热半群光滑化估计，改进了Tayachi20中利用弱Lebesgue和Lorentz空间证明的既有结果。

热核及树上热方程解的长时间渐进行为

Long-time asymptotics for the heat kernel and for heat equation solutions on homogeneous trees

研究同构树上热核及热方程解的长时间渐进行为，推导热核渐进行为并证明解在l^p范数下的分解性质，揭示图几何对热扩散的影响。