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Jianchun Chu

Publications and source records attributed to Jianchun Chu.

At least 37 records · Page 2Linked to original sources
arXiv2022

Kähler manifolds and mixed curvature

In this work we consider compact Kähler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef. Moreover, if the curvature is negative at some point, then the manifold is projective with canonical line bundle being big and nef. If in addition the curvature is negative, then the canonical line bundle is ample. As an application, we answer a question of Ni concerning manifolds with negative $k$-Ricci curvature and generalize a result of Wu-Yau and Diverio-Trapani to the conformally Kähler case. We also show that the compact Kähler manifold is projective and simply connected if the mixed curvature is positive.

math.DG↗
arXiv2022

Ricci-Deturck flow from rough metrics and applications

Motivated by the recent work of Lamm and Simon, in this work we study the short-time existence theory of Ricci-Deturck flow starting from rough metrics which are bi-Lipschitz and have small local scaling invariant gradient concentration. As applications, we use it to study several stability problems related to the scalar curvature.

math.DG↗
arXiv2022

Kähler tori with almost non-negative scalar curvature

Motivated by the torus stability problem, in this work we study Kähler metrics with almost non-negative scalar curvature on complex torus. We prove that after passing to a subsequence, non-collapsing sequence of Kähler metrics with almost non-negative scalar curvature will converge to flat torus weakly.

math.DG↗
arXiv2022

Lelong Numbers of $m$-Subharmonic Functions Along Submanifolds

We study the possible singularities of an $m$-subharmonic function $φ$ along a complex submanifold $V$ of a compact Kähler manifold, finding a maximal rate of growth for $φ$ which depends only on $m$ and $k$, the codimension of $V$. When $k < m$, we show that $φ$ has at worst log poles along $V$, and that the strength of these poles is moveover constant along $V$. This can be thought of as an analogue of Siu's theorem.

math.CV↗
arXiv2022

Monotonicity of the $p$-Green functions

On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can be viewed as a generalization of the recent result by Munteanu-Wang \cite{MunteanuWang2021} in the case of $p=2$. No smoothness assumption is made on the $p$-Green function when $1<p\leq 2$. Several rigidity results are also proven.

math.AP↗
arXiv2021

Fully non-linear elliptic equations on compact almost Hermitian manifolds

In this paper, we establish a priori estimates for solutions of a general class of fully non-linear equations on compact almost Hermitian manifolds. As an application, we solve the complex Hessian equation and the Monge--Ampère equation for $(n-1)$-plurisubharmonic equations in the almost Hermitian setting.

math.AP↗
arXiv2021

Hypercritical deformed Hermitian-Yang-Mills equation

In this work, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold. We introduce the notions of coerciveness and properness of the $\mathcal{J}$-functional on the space of almost calibrated $(1,1)$-forms and show that they are both equivalent to the existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.

math.DG↗
arXiv2021

Fully non-linear degenerate elliptic equations in complex geometry

We derive an a priori real Hessian estimate for solutions of a large family of geometric fully non-linear elliptic equations on compact Hermitian manifolds, which is independent of a lower bound for the right-hand side function. This improves on the estimates of Székelyhidi and additionally applies to elliptic equations with a degenerate right-hand side. As an application, we establish the optimal $C^{1,1}$ regularity of envelopes of $(θ, m)$-subharmonic functions on compact Hermitian manifolds.

math.DG↗
arXiv2021

A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills equation

The deformed Hermitian-Yang-Mills equation is a complex Hessian equation on compact Kähler manifolds that corresponds to the special Lagrangian equation in the context of the Strominger-Yau-Zaslow mirror symmetry. Recently, Chen proved that the existence of the solution is equivalent to a uniform stability condition in terms of holomorphic intersection numbers along test families. In this paper, we establish an analogous stability result not involving a uniform constant in accordance with a recent work on the $J$-equation by Song, which makes further progress toward Collins-Jacob-Yau's original conjecture in the supercritical phase case. In particular, we confirm this conjecture for projective manifolds in the supercritical phase case.

math.DG↗
arXiv2021

Conformal tori with almost non-negative scalar curvature

In this work, we consider sequence of metrics with almost non-negative scalar curvature on torus. We show that if the sequence is uniformly conformal to another sequence of metrics with uniformly controlled geometry, then it converges to a flat metric in the volume preserving intrinsic flat sense, $L^{p}$ sense and the measured Gromov-Hausdorff sense.

math.DG↗
arXiv2021

On the Hölder estimate of Kähler-Ricci flow

In this work, we study the H\"older regularity of the Kähler- Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. By adapting the method in the work of Hein-Tosatti on collapsing Calabi-Yau metrics, we obtain a uniform spatial Hölder estimate of the Kähler-Ricci flow for all time.

math.DG↗
arXiv2020

$C^{1,1}$ regularity of geodesics of singular Kähler metrics

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on Kähler manifolds away from the non-Kähler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of Kähler metrics on compact Kähler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge-Ampère equation that does not require strict positivity of the reference form near the boundary. We also discuss the case of some special geodesic rays.

math.DG↗
arXiv2020

The space of almost calibrated $(1,1)$ forms on a compact Kähler manifold

The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact Kähler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second author and Yau, and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon. This paper initiates the study of the geometry of $\mathcal{H}$. We show that $\mathcal{H}$ is an infinite dimensional Riemannian manifold with non-positive sectional curvature. In the hypercritical phase case we show that $\mathcal{H}$ has a well-defined metric structure, and that its completion is a ${\rm CAT}(0)$ geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case $\mathcal{H}$ admits $C^{1,1}$ geodesics, improving a result of the second author and Yau. Using results of Darvas-Lempert we show that this result is sharp.

math.DG↗
arXiv2018

$C^{1,1}$ regularity of degenerate complex Monge-Ampère equations and some applications

In this paper, we prove a $C^{1,1}$ estimate for solutions of complex Monge-Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show existence of $C^{1,1}$ solutions to the degenerate Monge-Ampère equations, the corresponding Dirichlet problems and the singular Monge-Ampère equations. We also study the singularities of the pluricomplex Green's function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge-Ampère type equations. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of Sasakian metrics.

math.AP↗
arXiv2018

The Fu-Yau equation on compact astheno-Kähler manifolds

In this paper, we study the Fu-Yau equation on compact Hermitian manifolds and prove the existence of solutions of equation on astheno-Kähler manifolds. We also prove the uniqueness of solutions of Fu-Yau equation when the slope parameter $α$ is negative.

math.DG↗