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Yalong Shi

Publications and source records attributed to Yalong Shi.

14 recordsLinked to original sources
arXiv2025

Green function rigidity and the mass of hypersurfaces under inversion

This is a sequel to arXiv:2401.02087. We prove the Green function rigidity conjecture in arXiv:2401.02087 for conformal Laplacian in dimension $n\geq 3$. For the Paneitz operator, we prove the Green function rigidity conjecture when $n\neq 4k+2, k\geq 2$. Important ingredients in our proof are the positive mass theorem and the positive energy theorem for Paneitz operator. As a byproduct, we also obtain a new formula for the ADM mass of an asymptotically flat hypersurface that allows for a non-entire graph.

math.DG↗
arXiv2024

CscK metrics near the canonical class

Let $X$ be a Kähler manifold with semi-ample canonical bundle $K_X$. It is proved by Jian-Shi-Song that for any Kähler class $γ$, there exists $δ>0$ such that for all $t\in (0, δ)$ there exists a unique cscK metric $g_t$ in $K_X+ t γ$. In this paper, we prove that $\{ (X, g_t) \}_{ t\in (0, δ)} $ have uniformly bounded Kähler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov-Hausdorff sense.

math.DG↗
arXiv2024

Green functions for GJMS operators on spheres, Gegenbauer polynomials and rigidity theorems

We derive explicit representation formulae of Green functions for GJMS operators on $n$-spheres, including the fractional ones. These formulae have natural geometric interpretations concerning the extrinsic geometry of the round sphere. Conversely, we discover that this special feature uniquely characterizes spheres among closed embedded hypersurfaces in $\mathbb{R}^{n+1}$. Furthermore, for $n=3,4,5$ we prove a strong rigidity theorem for Green functions of hypersurfaces in $\mathbb{R}^{n+1}$ using the Positive Mass Theorem.

math.DG↗
arXiv2021

Constant scalar curvature Kaehler metrics on ramified Galois coverings

We give sufficient conditions for the existence of Kaehler-Einstein and constant scalar curvature Kaehler (cscK) metrics on finite ramified Galois coverings of a cscK manifold in terms of cohomological conditions on the Kaehler classes and the branching divisor. This result generalizes previous work on Kaehler-Einstein metrics by Li-Sun [Comm. Math. Phys. 2014], and extends Chen-Cheng's existence results for cscK metrics in [J. Amer. Math. Soc. 2021].

math.DG↗
arXiv2014

A criterion for the properness of the K-energy in a general Kahler class

In this paper, we give a criterion for the properness of the K-energy in a general Kahler class of a compact Kahler manifold by using Song-Weinkove's result. As applications, we give some Kahler classes on $\mathbb{C}\mathbb{P}^2\#3\overline {\mathbb{C}\mathbb{P}^2}$ and $\mathbb{C}\mathbb{P}^2\#8\overline {\mathbb{C}\mathbb{P}^2}$ in which the K-energy is proper. Finally, we prove Song-Weinkove's result on the existence of critical points of $\hat J$ functional by the continuity method.

math.DG↗
arXiv2009

On the α-Invariants of Cubic Surfaces with Eckardt Points

In this paper, we show that the α_{m,2}-invariant of a smooth cubic surface with Eckardt points is strictly bigger than 2/3. This can be used to simplify Tian's original proof of the existence of Kaehler-Einstein metrics on such manifolds. We also sketch the computations on cubic surfaces with one ordinary double points, and outline the analytic difficulties to prove the existence of orbifold Kaehler-Einstein metrics.

math.AG↗