arXiv subjects

Xi Sisi Shen

Publications and source records attributed to Xi Sisi Shen.

10 recordsLinked to original sources

arXiv2026

Coupled continuity equations for constant scalar curvature Kähler metrics

Inspired by a parabolic system of Li-Yuan-Zhang and the continuity equation of La Nave-Tian, we study a system of elliptic equations for a Kähler metric $ω$ and a closed $(1, 1)$-form $α$. Assuming a uniform estimate for $ω$, we prove higher order estimates and smooth convergence to a cscK metric coupled to a harmonic $(1, 1)$-form. A simplification of the system is used to recover existence results for Kähler-Einstein metrics when $c_1(X) < 0$. On Riemann surfaces with genus at least $2$, we show smooth convergence to the unique Kähler-Einstein metric from a large class of initial data.

math.DG↗

arXiv2024

Non-preservation of $α$-concavity for the porous medium equation in higher dimensions

In this short note, we prove that $α$-concavity of the pressure is not preserved for the porous medium equation in dimensions $n=3$ and higher for any $α\in [0,1]\backslash \{\frac{1}{2}\}$. Together with the result of Chau-Weinkove for $n=2$, this fully resolves an open problem posed by Vásquez on whether pressure concavity is preserved in general for the porous medium equation.

math.AP↗

arXiv2023

The continuity equation for Hermitian metrics: Calabi estimates, Chern scalar curvature and Oeljeklaus-Toma manifolds

We prove local Calabi and higher order estimates for solutions to the continuity equation introduced by La Nave-Tian and extended to Hermitian metrics by Sherman-Weinkove. We apply the estimates to show that on a compact complex manifold the Chern scalar curvature of a solution must blow up at a finite-time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus-Toma manifolds we prove Gromov-Hausdorff and smooth convergence of the metric to a particular non-negative $(1,1)$-form as $t\to\infty$.

math.DG↗

arXiv2022

The continuity equation on Hopf and Inoue surfaces

We study the continuity equation of La Nave-Tian, extended to the Hermitian setting by Sherman-Weinkove, on Hopf and Inoue surfaces. We prove a priori estimates for solutions in both cases, and Gromov-Hausdorff convergence of Inoue surfaces to a circle.

math.DG↗

arXiv2022

Canonical almost-Kähler metrics dual to general plane-fronted wave Lorentzian metrics

In the compact setting, Aazami and Ream \cite{Aazami:2022th} proved that Riemannian metrics dual to a class of Lorentzian metrics, called (compact) general plane-fronted waves, are almost-Kähler. In this note, we explain how to construct extremal and second-Chern-Einstein non-Kähler almost-Kähler metrics dual to those general plane-fronted waves.

math.DG↗

arXiv2022

A Chern-Calabi flow on Hermitian manifolds

We study an analogue of the Calabi flow in the non-Kähler setting for compact Hermitian manifolds with vanishing first Bott-Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern-Ricci-flat metric in the $\partial\bar{\partial}$-class of the initial metric.

math.DG↗

arXiv2020

Estimates for metrics of constant Chern scalar curvature

We prove a priori estimates for constant Chern scalar curvature metrics on a compact complex manifold conditional on an upper bound on the entropy, extending a recent result by Chen-Cheng in the Kähler setting.

math.DG↗

arXiv2020

The Kähler-Ricci flow, holomorphic vector fields and Fano bundles

We study the behavior of the Kähler-Ricci flow on compact manifolds developing finite-time singularities, in particular, when the flow contracts exceptional divisors or collapses the Fano fibers of a holomorphic fiber bundle. We present a technique using holomorphic vector fields to prove estimates related to the work of Song-Weinkove and Fu-Zhang.

math.DG↗

arXiv2016

On Parallel Transport in Wasserstein Space

In this short note, we would like to give a construction of parallel transport for tangent cones lying in the interior of a geodesic in Wasserstein space. We give a complete proof for the linear part of the tangent space, and show that a construction for the full tangent cones follows from some natural lemmas on Wasserstein space. It can easily be shown that our construction is equivalent to those used in the previous literature on this subject.

math.DG↗

arXiv2013

A Rigidity Theorem for Hypersurfaces in Higher Dimensional Space Forms

The classical Cohn-Vossen theorem states that two isometric compact convex surfaces in $\mathbb{R}^{3}$ are congruent. In this short note, we generalize the classical Cohn-Vossen Theorem to higher dimensional surfaces in space form $N^{n+1}(K)$ for $n\ge 2$.

math.DG↗