Papers

In this paper, we prove that, under suitable conditions on the generating profile, every connected, compact embedded minimal capillary hypersurface supported on a one-ended rotational hypersurface in the Euclidean space is a horizontal slice. We also construct a smooth one-ended rotational support carrying a non-horizontal planar capillary $n$-ball, showing that the main slope condition cannot in general be omitted. For obtuse contact angles, we further obtain a catenoid-band classification under a reverse profile inequality.

We prove a vanishing theorem for the local Betti numbers of closed, $n$-dimensional Riemannian manifolds with a lower volume bound, an upper diameter bound, and a lower intermediate Ricci curvature bound. This generalizes and interpolates between known results under lower sectional and Ricci curvature bounds. Moreover, the methods extend to open manifolds, yielding a corresponding vanishing theorem for the global Betti numbers.

We prove that the curvature equation $\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{M}})=\sum_{j=1}^n \alpha_j E_j(\tau_{\mathcal{N}})$ yields uniqueness up to translation for any two closed $C^2_+$ hypersurfaces $\mathcal{M}, \mathcal{N}\hookrightarrow\mathbb{R}^{n+1}$ whenever $(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}_{\geq 0}^n\setminus\{0\}$ is log-concave and has no internal zeros. This gives an affirmative answer to a uniqueness question posed by Firey in the $C^2_+$ class.

We determine the sharp measure contraction exponents of two families of Grushin-type metric measure spaces. The radial Grushin space $\mathbb{G}^{n+m}$ is $\mathbb{R}^{n}\times\mathbb{R}^{m}$, equipped with Lebesgue measure and generated by $X_i=\partial_{x_i}$ and $Y_j=|x|\partial_{y_j}$, for $1\leq i\leq n$ and $1\leq j\leq m$. We prove that $\mathbb{G}^{n+m}$ satisfies $\operatorname{MCP}(K,N)$ if and only if $N\geq n+4m$ and $K\leq 0$. We also show that, for $\alpha\geq1$, the $\alpha$-Grushin plane generated by $X=\partial_x$ and $Y_\alpha=|x|^\alpha\partial_y$ satisfies $\operatorname{MCP}(K,N)$ if and only if $K\leq0$ and $N\geq N_\alpha$, where \[ N_\alpha := 1+\max_{L>1} \frac{(2\alpha+1)L}{(L-1)^{2\alpha+1}+1}. \] This resolves the conjecture posed in arXiv:2010.16350 and, for integer $\alpha\geq2$, provides the first examples of real-analytic sub-Riemannian structures with noninteger curvature exponent. Both results recover the known curvature exponent $5$ of the classical Grushin plane, corresponding respectively to $n=m=1$ and $\alpha=1$.

We show that on any closed orientable real-analytic Riemannian surface, any smoothly immersed closed immortal curve-shortening flow converges to a unique non-constant closed geodesic.

We construct a smooth complete metric on $\R^3$ with strictly positive Ricci curvature and $\displaystyle \lim_{s\rightarrow\infty}s^{-1}\int_{B_p(s)}\Sc\dd\mu= \infty$. This gives a negative answer to Shing-Tung Yau's question on the asymptotic scalar-curvature integral. We also give counterexamples to Naber's local Shing-Tung Yau conjecture on compact manifolds. The noncompact example has zero asymptotic volume ratio, and the compact counterexamples have volumes tending to zero.

In this paper, we prove several results concerning the function theory of complete Hermitian manifolds with vanishing Chern curvature, which we refer to as complete Chern-flat manifolds. First, we prove that any complete simply connected Chern-flat manifold is elliptic in the sense of Gromov. Moreover, it is biholomorphic to a complex Lie group if the torsion has polynomial growth. Next, we obtain sharp dimension estimates for holomorphic functions of polynomial growth on complete Chern-flat manifolds, with no growth assumption on the torsion. Finally, we prove that any complete Chern-flat manifold admitting polynomial-growth holomorphic functions that form a local holomorphic coordinate system at some point is biholomorphic to complex Euclidean space. These results generalize our previous work, in which sublinear growth of the torsion was assumed. The new analytic tool is a version of Cauchy estimates along orbits of holomorphic flows generated by parallel unitary frames on such manifolds.

This paper establishes a convergence theory for a continuous domain decomposition method for elliptic equations on manifolds. This method originated in \cite{lions2} in the setting of Euclidean domains and was later adapted and generalized to manifolds by \cite{qin_wang_wang}. Although its convergence was known, whether the convergence is geometric has remained open. We prove its geometric convergence and provide various estimates on the convergence rate.

We show that every area-preserving Lipschitz map between metric surfaces distorts the length of almost every curve by at most a multiplicative factor, answering a question posed by the first author and Ntalampekos. We give two different proofs. The first is entirely intrinsic and extends to higher dimensions under additional assumptions. The second relies on non-smooth uniformization theory and yields the conclusion that the family of curves intersecting the purely 2-unrectifiable part of a metric surface in a set of positive length is exceptional.

Let $(M^3,g)$ be a complete Riemannian manifold diffeomorphic to $\R^3\setminus\{0\}$, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass $m_+$. For each $p\in(1,3)$, define $c_{O,p}$ as the infimum of the Schwarzschild-normalized $p$-capacity over outward-minimizing finite-perimeter boundaries separating the two ends. We prove that $m_+\ge c_{O,p}$ whenever $A(g)>0$, where $A(g)$ is the infimum of the areas of boundaries separating the two ends. Equality at a single exponent produces a least-area horizon, forces its exterior to be the Schwarzschild exterior of mass $m_+$, and yields equality at every exponent. In the equality case, if the second end is also asymptotically flat, its mass satisfies $m_-\ge m_+$, with equality precisely for the two-sided spatial Schwarzschild manifold. We also show that a strict gap between $c_{O,p}$ and the unconstrained capacity infimum $c_{M,p}$ detects a horizon.

Let $X$ be a compact Riemann surface of genus $g\geq2$, and let $\M$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $$ \Vol_{g_{L^2}}B_{L^2}(p,R) =\frac{2^{4g-3}\pi^{9g-9}}{(3g-3)!}\,R^{6g-6}+o(R^{6g-6}) $$ for every $p\in\M$. We also determine the leading asymptotics of Hamiltonian sublevel volumes and exponentially weighted volumes. The proof combines homogeneity of the hyperkähler volume form with metric asymptotics on the regular Hitchin locus. Symplectic reduction and the Prym polarization evaluate the coefficient, which is independently recovered by equivariant localization.

In this article, we study multiply warped products with a one-dimensional base, harmonic Weyl tensor and zero radial Weyl curvature. We prove that these manifolds have at most three fibers and describe their local geometry. The proof uses a conformal change which makes the warping functions quadratic polynomials, together with the compatibility equations imposed by the Ricci tensor. The resulting metrics are either obtained from locally conformally flat multiply warped products by replacing their space-form fibers with Einstein manifolds, or belong to a three-fiber family with equal fiber dimensions and positive Einstein constants. Examples show that the bound is sharp and that neither Weyl condition implies the other. We also discuss the stronger restrictions arising from soliton-type equations.

In this paper, we establish the decomposition and product theory of $\Gamma$-symmetric spaces, where $\Gamma$ is isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$. These results lead to the local classification of semisimple Riemannian $\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric spaces of nontrivial type. Furthermore, we identify notable phenomena specific to $\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric spaces: the existence of distinct product structures and that of an example with the trivial infinitesimal isotropy representation.

In this paper, based on the warped product structures determined by the Obata-type equations with Robin boundary condition, we establish some rigidity results for compact manifolds with smooth boundary under appropriate Bakry-Émery Ricci curvature conditions and some other assumptions.

Kimberling's Problem 10 asks for a simple closed curve of prescribed length $L$ (in particular, $L=4\pi$) on the unit sphere minimizing the mean geodesic distance $\mathcal{J}$ from a point of the sphere to the curve. For a positive integer $n$, put $\vartheta_{n}=\pi/(2n)$ and $L_{n}=2\pi/\sin\vartheta_{n}$. We show that the minimum of $\mathcal{J}$ over rectifiable simple closed curves of length at most $L_{n}$ equals $\vartheta_{n}-\tan(\vartheta_{n}/2)$, that it is attained only by curves of length exactly $L_{n}$, and that the sphere-filling ropes $\beta^{n,k}$ of Gerlach and von der Mosel attain it. Kimberling's case is $n=3$: at $L=4\pi$ the minimum is $\pi/6+\sqrt{3}-2=0.255649\ldots$, attained by an explicit six-arc curve and by its mirror image. For $L\le2\pi$ we determine $J(L)$, the infimum of $\mathcal{J}$ over curves of length $L$, exactly: it equals $\pi/2-L/(2\pi)$, attained precisely by the circles of length $L$. At the lengths $L_{n}$ we do not classify all minimizers, but show that every one of them bisects the sphere into two disks of area $2\pi$ and inradius $\vartheta_{n}$ whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for $n=1$, and the $\beta^{n,k}$ are, up to congruence, the only ones of thickness at least $\sin\vartheta_{n}$. For arbitrary $L$ the function $J$ is nonincreasing, and together with the above this brackets it between two explicit values.

In this paper we prove that simple Finsler manifolds are conformally stable. Given a simple Finsler manifold and a class of conformal factors, we characterize the singularity of their induced boundary distance functions which allows us to define an appropiate $H^2$ norm. We then obtain a stability estimate with respect to this Sobolev norm and the $L^2$ norm on the class of the conformal factors. To prove the main theorems, we adapt the classical integration by parts technique employed by Mukhometov \cite{Muhometov} to the Finsler setting.

The magnetic field strength is a central design object in the theory of magnetic confinement of plasma. Desirable confinement properties such as quasi-symmetry and omnigenity are characterized by special forms of the field strength when it is given in the particular coordinate system called “Boozer coordinates." We ask a basic realizability question: which functions of Boozer coordinates can arise as the magnitude of a magnetic field that is written in Boozer coordinates? Using exterior differential systems and the Cartan-Kähler theorem, we show that every positive analytic function is locally realizable. Thus there is no local analytic obstruction to prescribing the field strength in Boozer coordinates; though further restrictions might arise from global geometric or additional physical constraints.

We develop an area-constrained Willmore theory for closed spacelike surfaces in four-dimensional Lorentzian manifolds. Since the Euler–Lagrange equation is normal-bundle-valued, we introduce directional criticality along a prescribed normal direction and identify a class of connection-compatible directions determined by the normal connection. Our main result is the sharp inequality $2\lambda|\Sigma|+\int_\Sigma |\vec H|^2,d\mu\leq 8\pi\chi(\Sigma)$ for criticality in a connection-compatible spacelike direction under an Einstein-tensor sign condition. Under the dominant energy condition and $\lambda\geq0$, this yields $\int_\Sigma |\vec H|^2,d\mu\leq16\pi$. Equality is rigid: $\Sigma$ is round and satisfies Minkowski causal-diamond rigidity. We also prove an analogous sharp inequality for null critical directions. These results give positivity, rigidity, stationarity, and infinitesimal monotonicity statements for the Hawking quasi-local energy. A hypersurface formulation recovers the previously studied Hawking-surface problem as criticality in the spatial normal direction. Explicit examples in spherical symmetry, Kerr, and FLRW spacetimes distinguish directional from full criticality and show the necessity of the sign assumptions in the rigidity theory.

Let $M$ be a complete open $5$-manifold with nonnegative Ricci curvature. If its universal cover $\tilde M$ has Euclidean volume growth, then $\pi_1(M)$ is finitely generated and virtually $\mathbb Z^k$ for some $0\le k\le4$. This confirms a conjecture of Pan and Rong \cite{PR18} in dimension five; see also \cite{BNS25,BrNaSe25}. In dimension six, under the same volume growth assumption, we also obtain a quantitative virtual abelianness result without assuming finite generation of the fundamental group.

We prove the existence of arbitrarily high-dimensional families of minimal surfaces of any prescribed genus and conformal structure in even-dimensional round spheres. More precisely, let $n\geq2$ and let $\Sigma$ be any closed Riemann surface of genus $g$. We construct a sequence of degrees $d_\ell\to+\infty$ such that, for every $\ell$, there exists a complex manifold of complex dimension $2d_\ell+n^2(1-g)$ consisting of linearly full branched superminimal immersions of $\Sigma$ into $\mathbb S^{2n}$ of degree $d_\ell$. In particular, this yields parametrized families of linearly full branched minimal immersions whose dimensions tend to infinity.

We construct new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the conifold $\mathcal{Z} = \{z_1^2 + z_2^2 + z_3^2 + z_4^2 = 0\} \subset \mathbb{C}^4$. These metrics have tangent cone $\mathbb{C} \times (\mathbb{C}^2 / \mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a Calabi-Yau metric on $\mathcal{Z}$ with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point and tangent cone at infinity $\mathbb{C} \times (\mathbb{C}^2/\mathbb{Z}_2)$, thereby providing a new metric realisation of the Atiyah flop.

We construct a numerically flat holomorphic vector bundle of rank two on a compact complex threefold satisfying the ordinary $\partial\bar\partial$-lemma and prove that it admits no holomorphic connection. This gives, in particular, a negative answer to a question posed by Cao–Deng–Matsumura. In contrast, for compact simply connected complex manifolds $X$, we prove that the answer is affirmative if $H^1(X,\mathcal O_X)=0$, and hence if the Frölicher spectral sequence of $X$ degenerates at $E_1$.

We establish spacetime counterparts of fill-in inequalities for spin initial data sets satisfying the dominant energy condition in dimensions three through seven, controlling the minimum and the integral of the norm of the spacetime mean curvature vector by intrinsic boundary data. The proof relies on a Reilly–MIT estimate for complete spin manifolds with compact boundary. As further consequences, under suitable nonnegativity assumptions, Gromov's hyperspherical-radius fill-in inequality and Bär's recent total mean-curvature estimate extend to complete, possibly noncompact spin manifolds with compact boundary.

In this article we provide a systematic reduction to static spherical symmetry for (classical) gauge theoretic models with structure group SU(2) x U(1), coupled to gravity. In particular, we derive the most general static spherically symmetric ansatz for the Einstein-Yang- Mills-Higgs-Dirac-Yukawa equation system. This model is reminiscent of the lepton sector of the Standard Model coupled to gravity. The resulting total system of ordinary differential equations is rather large, so that finding global solutions is non-trivial. For this reason, an open-source interactive solver GUI written in C++ is provided together with the article.

We characterize equality in the mass-capacity inequality for asymptotically flat manifolds with arbitrary ends in every dimension \(n\ge 3\). If \(u\) is the capacity minimizer, then the equality forces \((M,u^{4/(n-2)}g)\) to be isometric to \((\mathbb R^n\setminus S, g_{\mathrm{Euc}})\) for a compact set \(S\) satisfying \(\operatorname{dim}_{\mathcal H}S\le (n-2)/2\). A key ingredient is that conformal changes by positive harmonic functions transform nonnegative Ricci curvature into nonnegative Bakry–Émery Ricci curvature of effective dimension \(4-n\).

We study radial Kähler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a Kähler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini-Study metric, the flat metric, or, in complex dimension two, generalized Burns-Simanca metrics. We also prove that every radial projectively induced Kähler-Einstein metric has constant holomorphic sectional curvature and is therefore a Fubini-Study, flat, or complex hyperbolic metric. Finally, we show that a radial infinitely projectively induced extremal Kähler metric has unbounded maximal radial domain if and only if it is scalar-flat.

Let $M^n$ be a closed oriented manifold of even dimension $n=2m\ge2$, equipped with a smooth metric $g_1$ and a Riemannian metric $g_2$. We compute the Wodzicki residue of $D_{g_1}^2D_{g_2}^{-n}$ on the common exterior bundle, where $D_{g_r}=d+\delta_{g_r}$. This is the noncommutative integral of $D_{g_1}^2$ relative to the reference operator $D_{g_2}$. Its local density involves the curvatures of the two metrics and the difference of their Levi–Civita connections. Integration by parts gives a closed-manifold formula without explicit derivatives of the connection difference. For compact manifolds with boundary and $n\ge4$, we assume that both metrics are Riemannian and satisfy $g_r=h_r(x_n)^{-1}g^{\partial M}+dx_n^2$ near the boundary, with $h_r(0)=1$. We compute the noncommutative residues of two factorizations involving even and odd powers of $D_{g_2}$. Their interior contributions coincide, whereas their boundary terms are explicit multiples of the integral of $K_{g_1}-K_{g_2}$, with the coefficient for the even factorization twice that for the odd one. Here $K_{g_r}$ is the trace of the second fundamental form with respect to the inward unit normal. The calculation uses a second-order residue formula and direct boundary symbol expansions. When the metrics coincide, the formulas reduce to the Hodge–de Rham Kastler–Kalau–Walze identity.

We study the local continuation of a steady vortex sheet under a prescribed deformation $g_s= e^{2\sigma_s}g_{\mathrm{e}}$ of a conformal metric on the unit disk. Representing the moving sheet by a normal graph, we formulate the streamline and Bernoulli conditions as a nonlinear residual whose two components have different Sobolev orders. If the two one-sided tangential velocities of the reference sheet do not vanish simultaneously, the derivative of the unnormalized residual is Fredholm of index two. Fixing the mean normal displacement and the total circulation gives an index-zero problem; when the normalization differential restricted to the unnormalized kernel is onto $\mathbb{R}^2$, we obtain a parameter-dependent Lyapunov-Schmidt reduction. This yields a locally unique normalized branch in the nondegenerate case and explicit first- and second-order necessary conditions in the degenerate case. For a rotationally symmetric reference metric and a concentric circular sheet, the linearized problem separates into Fourier blocks and only finitely many modes can be resonant. In the constant-curvature disk family, every mode $k\geq2$ is nonresonant, whereas the first mode produces a two-dimensional reduced problem. The two first-mode kernel profiles are normal traces of ambient Killing fields, but they do not generate fixed-boundary symmetries or, by themselves, a nonconcentric branch.

We prove that every compact connected Riemannian isometric filling $M$ of a circle of length $2\pi$ satisfies $\operatorname{Area}(M) \geq \frac{14\zeta(3)}{\pi} \approx 5.35677$, regardless of orientability or topological types. Our new approach uses the odd Fourier coefficients of the distance functions from boundary points. For orientable fillings, we use a cubic resonant perturbation to obtain $\operatorname{Area}(M)>5.40154$.

We classify compact Hermitian manifolds with flat Levi-Civita connection. This is equivalent to the classification of all orthogonal complex structures on flat tori. It generalized the work of Khan, Yang, and Zheng in 2017 where they solved the case in complex dimension three.