A Fourth-Order Positivity Preserving Geometric Flow

Positivity preserving along a flow over projective bundle

In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the …

$L^\infty$-正性保持性质与随机完备性

The $L^\infty$-positivity preserving property and stochastic completeness

本文研究了 Riemannian 流形的 $L^\infty$-正性保持性质，证明其等价于随机完备性（即最小热核保持概率），并通过分布解的单调逼近结果实现该结论。

$L^{p}$ Positivity Preserving and a conjecture by M. Braverman, O. Milatovic and M. Shubin

In this paper we prove that a complete Riemannian manifold is $L^p$-positivity preserving for any $p\in(1,\infty)$. This means that any $L^p$ function which solves $(-Δ+ 1)u\ge 0$ in the sense of distributions is necessarily non-negative. In particular, the case $p=2$ of our result answers in the affermative a conjecture formulated by M. Braverman, O. Milatovic and M. Shubin in 2002. The two main ingredients are a new a-priori regularity result for positive subharmonic distributions, which in turn permits to prove a Liouville type theorem, and a Brezis-Kato inequality on Riemannian manifolds. …

一种中心等距几何中的四阶面积保持曲线流

A fourth-order area-preserving curve flow in centro-equiaffine geometry

本文基于Guan和Li(2015)的工作，引入一种通过仿射Minkowski公式定义的四阶中心等距不变曲线流，证明在无初始曲线小性假设下，流在长时间内存在，且曲线面积保持并最终光滑收敛到圆。

The Geometry of Fourth Order Differential Operator

In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.