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Jianchun Chu

Publications and source records attributed to Jianchun Chu.

At least 19 recordsLinked to original sources
arXiv2026

Rigidity of positive mass theorem with fast metric decay

In this work, we consider metrics on Euclidean space with nonnegative scalar curvature and rapid decay at infinity. We show that, in dimensions four and higher, any such metric is necessarily flat if its decay rate exceeds that of the Schwarzschild metric. This complements recent works by Mazurowski-Yao and You-Zhang, thereby establishing Gromov's conjecture on the rigidity of the positive mass theorem under fast metric decay in all dimensions. Our method also extend naturally to weakly asymptotically flat manifolds.

math.DG↗
arXiv2026

Some variational problems for the complex Monge--Amp{è}re operator

We consider the Dirichlet problem for the complex Monge--Ampère equation on strongly pseudoconvex Kähler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth solutions in a number of natural circumstances, following work of Chou-Wang.

math.CV↗
arXiv2026

The rigidity of dimension estimate for holomorphic functions on Kähler manifolds

In this paper, we obtain the optimal rigidity of dimension estimate for holomorphic functions with polynomial growth on Kähler manifolds with non-negative holomorphic bisectional curvature. There is a specific gap between the largest and the second largest dimension. We also determine the optimal dimension that ensures the maximal volume growth which implies the manifold is biholomorphic to the complex Euclidean space.

math.DG↗
arXiv2025

Positive scalar curvature metrics and aspherical summands

We prove for $n\in\{3,4,5\}$ that the connected sum of a closed aspherical $n$-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result answers a question of Gromov. More generally, we generalize the partial classification result of Chodosh, Li, and Liokumovich to the non-compact domination case with our newly-developed technique. Our result unifies all previous results of this type, and confirms the validity of Gromov's non-compact domination conjecture for closed aspherical manifolds of dimensions 3, 4, and 5.

math.DG↗
arXiv2025

Interior control for surfaces with positive scalar curvature and its application

Let $M^{n}$, $n\in\{3,4,5\}$, be a closed aspherical $n$-manifold and $S\subset M$ a subset consisting of disjoint incompressible embedded closed aspherical submanifolds (possibly with different dimensions). When $n =3,4$, we show that $M\setminus S$ cannot admit any complete metric with positive scalar curvature. When $n=5$, we obtain the same result when $S$ contains a submanifold of codimension 1 or 2. The key ingredient is a new interior control for the extrinsic diameter of surfaces with positive scalar curvature.

math.DG↗
arXiv2024

The rigidity of eigenvalues on Kähler manifolds with positive Ricci lower bound

In this work, optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound are established. More precisely, for those Kähler manifolds whose first eigenvalue agrees with the Ricci lower bound, we show that the complex projective space is the only one with the largest multiplicity of the first eigenvalue. Moreover, there is a specific gap between the largest and the second largest multiplicity. In the Kähler--Einstein case, almost rigidity results for eigenvalues are also obtained.

math.DG↗
arXiv2024

Homological $n$-systole in $(n+1)$-manifolds and bi-Ricci curvature

In this paper, we prove an optimal systolic inequality and the corresponding rigidity in the equality case on closed manifolds with positive bi-Ricci curvature, which generalizes the work of Bray-Brendle-Neves. The proof is given in all dimensions based on the method of minimal surfaces under the Generic Regularity Hypothesis, which is known to be true up to dimension ten.

math.DG↗
arXiv2024

On Kähler manifolds with non-negative mixed curvature

In this work, we investigate compact Kähler manifolds with non-negative or quasi-positive mixed curvature coming from a linear combination of the Ricci and holomorphic sectional curvature, which covers various notions of curvature considered in the literature. Specifically, we prove a splitting theorem, analogous to the Cheeger-Gromoll splitting theorem, for complete Kähler manifolds with non-negative mixed curvature containing a line, and then establish a structure theorem for compact Kähler manifolds with non-negative mixed curvature. We also show that the Hodge numbers of compact Kähler manifolds with quasi-positive mixed curvature must vanish. Both results are based on the conformal perturbation method.

math.DG↗
arXiv2024

Llarull's theorem on punctured sphere with $L^\infty$ metric

The classical Llarull theorem states that a smooth metric on $n$-sphere cannot have scalar curvature no less than $n(n-1)$ and dominate the standard spherical metric at the same time unless it is the standard spherical metric. In this work, we prove that Llarull's rigidity theorem holds for $L^{\infty}$ metrics on spheres with finitely many points punctured. This is related to a question of Gromov.

math.DG↗
arXiv2024

The Eigenvalue Problem for the Complex Hessian Operator on $m$-Pseudoconvex Manifolds

We establish $C^{1,1}$-regularity and uniqueness of the first eigenfunction of the complex Hessian operator on strongly $m$-pseudoconvex manifolds, along with a variational formula for the first eigenvalue. From these results, we derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue.

math.CV↗
arXiv2023

Liouville theorem for a class of Hessian equations

In this paper, we study a general class of Hessian elliptic equations, including the Monge-Ampère equation, the $k$-Hessian equation and $p$-Monge-Ampère equations. We propose new additional condition on the solution and prove Liouville theorem under this assumption. We show that our general condition covers as special cases numerous sets of assumptions known in the literature which were tailored for specific equations. Thus we obtain a significant generalization of multiple isolated Liouville theorems and conditional interior estimates.

math.AP↗
arXiv2023

A non-spin method to the positive weighted mass theorem for weighted manifolds

In this paper, we investigate the weighted mass for weighted manifolds. By establishing a version of density theorem and generalizing Geroch conjecture in the setting of $P$-scalar curvature, we are able to prove the positive weighted mass theorem for weighted manifolds, which generalizes the result of Baldauf-Ozuch to non-spin manifolds.

math.DG↗
arXiv2022

Singular positive mass theorem with arbitrary ends

Motivated by the recent progress on positive mass theorem for asymptotically flat manifolds with arbitrary ends and the Gromov's definition of scalar curvature lower bound for continuous metrics, we start a program on the positive mass theorem for asymptotically flat manifolds with $C^0$ arbitrary ends. In this work as the first step, we establish the positive mass theorem of asymptotically flat manifolds with $C^0$ arbitrary ends when the metric is $W^{1,p}_{\mathrm{loc}}$ for some $p\in(n,\infty]$ and is smooth away from a non-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$. New techniques are developed to deal with non-compactness of the singular set.

math.DG↗
arXiv2022

Rigidity on non-negative intermediate curvature

In a recent work of Brendle-Hirsch-Johne, a notion of intermediate curvature was introduced to extend the classical non-existence theorem of positive scalar curvature on torus to product manifolds. In this work, we study the rigidity when the intermediate curvature is only non-negative when the ambient dimension is at most $5$.

math.DG↗
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↗