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Xumin Jiang

Publications and source records attributed to Xumin Jiang.

15 recordsLinked to original sources
arXiv2026

Mass Bounds for Confined Area-Minimizing Minimal Surfaces

We establish codimension-independent mass bounds for geometrically confined area-minimizing rectifiable currents. In Euclidean space, we combine the confined-volume doubling theorem of Colding--Minicozzi with a current-theoretic squashing argument. This gives an affirmative answer to Lin's interior mass-bound problem for every algebraic projection multiplicity $Q$: the interior mass is bounded by $C(n)Q$, without an a priori mass bound at a larger scale. This result yields a degree-one Bernstein-type rigidity result under sublinear confinement. For the hyperbolic application, we make the curvature modification of the fixed-scale argument needed in a thin tubular neighborhood of a totally geodesic copy of $\mathbb{H}^n$. Combining this auxiliary estimate with a localized squashing estimate removes the doubly exponential local mass-growth condition from the boundary regularity results in [13].

math.DG↗
arXiv2026

On the proof of Bray's conjecture

Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.

math.DG↗
arXiv2026

Bonnet-Myers type theorems for $Q$-curvature on four-manifolds

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. First, if the $Q$-curvature $Q_g\geq 6k^2$ and scalar curvature $R_g\geq -12k$ for some positive constant $k$, then $(M^4,g)$ is either Einstein with $Ric_g=-3kg$ or compact with $R_g\ge 12k$. As a corollary, the fundamental group $\pi_1(M^4)$ satisfies $|\pi_1(M^4)|\leq 16\pi^2/(\int_{M^4}Q_g d\mu_g),$ under the additional assumption $R_g>-12k$. Second, if the scalar curvature $R_g>0$ and $Q_g\geq \theta R_g$ for a positive constant $\theta$, then $M^4$ is compact and the diameter of $(M^4,g)$ is at most $4\pi/\sqrt{15\theta}$.

math.DG↗
arXiv2026

Asymptotics of high-codimensional area-minimizing currents in hyperbolic space

We investigate the asymptotic behavior of high-codimensional area-minimizing locally rectifiable currents in hyperbolic space, addressing a problem posed by F.H. Lin and establishing ``boundary regularity at infinity" results for such currents near their asymptotic boundaries under the standard Euclidean metric. Intrinsic obstructions to high-order regularity arise for odd-dimensional minimal surfaces, revealing a constraint dependent on the geometry of the asymptotic boundary. Our work advances the asymptotic theory of high-codimensional minimal surfaces in hyperbolic space.

math.DG↗
arXiv2024

A continuous cusp closing process for negative K\"ahler-Einstein metrics

We give an example of a family of smooth complex algebraic surfaces of degree $6$ in $\mathbb{CP}^3$ developing an isolated elliptic singularity. We show via a gluing construction that the unique K\"ahler-Einstein metrics of Ricci curvature $-1$ on these sextics develop a complex hyperbolic cusp in the limit, and that near the tip of the forming cusp a Tian-Yau gravitational instanton bubbles off.

math.DG↗
arXiv2023

The singular sets of degenerate and nonlocal elliptic equations on Poincar\'e-Einstein manifolds

The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincar\'e-Einstein manifolds. These operators generally reflect the correspondence between the Riemannian geometry of a complete Poincar\'e-Einstein manifold and the conformal geometry of its associated conformal infinity. In this setting, we develop the quantitative differentiation theory that includes quantitative stratification for the singular set and Minkowski type estimates for the (quantitatively) stratified singular sets. All these, together with a new $\epsilon$-regularity result for degenerate/singular elliptic operators on Poincar\'e-Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper provide a delicate synergy between the geometry of Poincar\'e-Einstein manifolds and the elliptic theory of associated degenerate elliptic operators.

math.DG↗
arXiv2021

Asymptotics of K\"ahler-Einstein metrics on complex hyperbolic cusps

Let $L$ be a negative holomorphic line bundle over an $(n-1)$-dimensional complex torus $D$. Let $h$ be a Hermitian metric on $L$ such that the curvature form of the dual Hermitian metric defines a flat K\"ahler metric on $D$. Then $h$ is unique up to scaling, and, for some closed tubular neighborhood $V$ of the zero section $D \subset L$, the form $\omega_h = -(n+1)i\partial\overline\partial\log(-{\log h})$ defines a complete K\"ahler-Einstein metric on $V \setminus D$ with ${\rm Ric}(\omega_h) = -\omega_h$. In fact, $\omega_h$ is complex hyperbolic, i.e., the holomorphic sectional curvature of $\omega_h$ is constant, and $\omega_h$ has the usual doubly-warped cusp structure familiar from complex hyperbolic geometry. In this paper, we prove that if $U$ is another closed tubular neighborhood of the zero section and if $\omega$ is a complete K\"ahler-Einstein metric with ${\rm Ric}(\omega) = -\omega$ on $U \setminus D$, then there exist a Hermitian metric $h$ as above and a $\delta \in \mathbb{R}^+$ such that $\omega - \omega_{h} = O(e^{-\delta\sqrt{-{\log h}}})$ to all orders with respect to $\omega_h$ as $h \to 0$. This rate is doubly exponential in the distance from a fixed point, and is sharp.

math.DG↗
arXiv2020

The Loewner-Nirenberg Problem in Cones

We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Due to the singularity of the spherical domains, extra cares are needed for the study of the global regularity of the eigenfunctions and solutions of the associated singular Dirichlet problem.

math.AP↗
arXiv2020

Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities

We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Key step is the study of the eigenvalues growth and eigenfunctions estimates.

math.AP↗
arXiv2018

Asymptotics and convergence for the complex Monge-Ampere equation

We study the asymptotics of complete Kaehler-Einstein metrics on strictly pseudoconvex domains in C^n and derive a convergence theorem for solutions to the corresponding Monge-Ampere equation. If only a portion of the boundary is analytic, the solutions satisfy Gevrey type estimates for tangential derivatives. A counterexample for the model linearized equation suggests that there is no local convergence theorem for the complex Monge-Ampere equation

math.AP↗
arXiv2018

Optimal Regularity of Constant Graphs in Hyperbolic Space

Inspired by [6, 7], we study the boundary regularity of constant curvature hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$, which have prescribed asymptotic boundary at infinity. Through constructing the boundary expansions of the solutions, we derive the optimal regularity of the solutions. Moreover, we obtain an equivalent condition that guarantees the smoothness of the solutions.

math.AP↗
arXiv2018

The convergence of boundary expansions and the analyticity of minimal surfaces in the hyperbolic space

We study expansions near the boundary of solutions to the Dirichlet problem for minimal graphs in the hyperbolic space and prove the local convergence of such expansions if the boundary is locally analytic. As a consequence, we prove a conjecture by F.-H. Lin that the minimal graph is analytic up to the boundary if the boundary is analytic and the minimal graph is smooth up to the boundary.

math.AP↗
arXiv2014

Boundary regularity of minimal graphs in the hyperbolic space

F.-H. Lin studied minimal graphs of the Dirichlet problem in the hyperbolic space and proved that any such minimal graph has the same global regularity as the boundary if the dimension of the minimal graph is even and that there is an obstacle to the higher regularity if the dimension is odd. We discuss the odd dimension case and study how the higher regularity is obstructed. We introduce the logarithm of the distance to the boundary as an additional independent self-variable and establish concise boundary regularity for the minimal graph.

math.AP↗