Papers

We prove a uniform scalar curvature bound, depending only on the dimension, for Kähler–Ricci shrinkers. We also prove that every noncompact Kähler–Ricci shrinker has a unique tangent cone at infinity, naturally homeomorphic to the base of its polarized Fano fibration.

We formulate and solve a $G$-equivariant analogue of the Seiberg-Witten equations on any irreducible Hermitian symmetric space $G/K$. We use a canonical $K$-invariant form to define a twisted Hodge operator generalising self-duality to higher dimensions. The resulting equivariant Seiberg-Witten equations arise, for a compact $G/K$, as conditions for the absolute minima of an energy functional. Their linear part has a representation theoretic interpretation: the same Weyl group data control the highest-weight realisation of the positive Dirac kernel in the compact case and its discrete-series realisation on the non-compact dual. This space of harmonic spinors collapses almost entirely by the quadratic conditions of the monopole equation. In the compact case, the equivariant moduli set, when non-empty, is a single gauge class, identifying with the $G$-invariant part of Bradlow's trichotomy for vortices, while in the non-compact case, every irreducible solution sits outside the discrete series.

We study overdetermined problems for normalized sum Hessian equations and their quotients in the space forms. The equations and boundary conditions imply admissibility and ellipticity. For quotient equations, we prove that the domain is a geodesic ball if it is strictly star-shaped, or if the Dirichlet value satisfies an upper bound. We also prove the corresponding result for sum Hessian equations without star-shapedness. The proofs use two auxiliary functions and a Rellich–Pohožaev identity.

We study spectral zeta densities associated with complex powers of principally scalar Laplace-type operators on pseudo-Riemannian scattering spaces of mixed signature and arbitrary dimension. The construction extends, beyond the even-dimensional Lorentzian case, the local analysis of Feynman complex powers developed by Dang and Wrochna. Namely, for $\varepsilon > 0$, we consider complex powers $(P\pm i\varepsilon)^{-\alpha}$ and prove that the diagonal trace densities admit meromorphic continuation. We relate their poles to local geometric invariants, in particular to the pseudo-Riemannian scalar curvature. Our proof combines uniform Feynman resolvent estimates with a direct analysis of the Hadamard parametrix in pseudo-Riemannian signature. Beyond the Lorentzian case, the use of causal convexity is replaced by a geometric no-return argument.

We prove a canonical quantization-commutes-with-reduction theorem for CR functions. Under the geometric hypotheses of Hsiao–Ma–Marinescu, if the zero moment level is nonempty and the strictly pseudoconvex reduction is globally CR embeddable of real dimension at least three, the CR Guillemin–Sternberg map is a bounded isomorphism at every real Sobolev order. The central new step is an algebraic and complex-analytic elimination of its finite-dimensional Fredholm defect: multiplicativity removes the kernel, while the conductor of the smooth range, integral dependence, and holomorphic removability produce global lifts. Conductor nonvanishing follows from the reduced Szegő singularity; we also prove it using global CR peak functions. The map is an isomorphism of Fréchet algebras, the reduction is connected, and the corresponding maps for holomorphic line bundles positive near the zero level are isomorphisms in every nonnegative tensor degree. Invariant holomorphic sections vanish in every negative degree; when the reduced base has positive complex dimension, the section maps are isomorphisms in every integer degree.

Let $S$ be a closed oriented surface of genus at least $2$. Let $X$ be a $(2,1)$-dimensional anti-de Sitter spacetime homeomorphic to $S\times[-1,1]$ with smooth, spacelike, strongly convex boundary. By Gauss' formula, the induced metric on each of the two components of $\partial X$ has curvature $K<-1$. In this paper we address the inverse problem. We show that, for any two smooth metrics $h_-, h_+$ on $S$ with curvature $K<-1$, there exists a unique, smooth anti-de Sitter metric on $X:=S\times [-1,1]$ with respect to which the boundary is locally strongly convex and has induced metric isotopic to $h_-$ on its past component and isotopic to $h_+$ on its future component. Existence has already been proven by Tamburelli. We use this result to solve affirmatively a conjecture of Mess: given any two measured laminations $l_-, l_+\in\mathcal{ML}_S$ that together fill $S$, there exists a unique $(2,1)$-dimensional, globally hyperbolic, maximal, Cauchy compact (GHMC) anti-de Sitter spacetime such that the measured bending laminations of the past and future boundary components of its convex core are respectively $l_-$ and $l_+$. Here, existence has already been proven by Bonsante–Schlenker.

The corpus callosum (CC) is a major white-matter structure and a well-established marker of brain aging, but most studies quantify it using scalar summaries that discard its boundary geometry. We present a Riemannian shape-space framework \cite{fletcher2013geodesic-regression} for analyzing age-related morphological change in the midsagittal CC, applied to the OASIS-1 cohort. Each contour is represented by $128$ landmarks and embedded into Kendall shape space, where translation, rotation, and scale are removed. We derive a multivariate geodesic regression with exact Riemannian gradients and use the fitted age-velocity field to localize age-related deformation to five anatomical sub-regions. In the cognitively normal cohort ($n = 252$), geodesic regression outperforms the Euclidean linear benchmark ($R^2 = 0.1355$ vs.\ $0.1216$). Regional energy is anterior-dominant: the Genu carries $46.1\%$ and the Anterior Body $23.3\%$ of total age-related shape change, together accounting for $\sim 69\%$ despite comprising only $\sim 41\%$ of landmarks. Signed projections confirm the ordering (Genu $r = 0.562$; Anterior Body $r = 0.507$). In contrast, age explains less than $0.5\%$ of shape variance in Alzheimer's disease ($n = 88$), indicating that the disease disrupts the healthy aging trajectory. A tangent-space classifier achieves an age-group AUC of $0.791$ from the 2D contour alone, exceeding a recent volumetric benchmark ($0.67$).

Let $X: M^n\to \mathbb S^{n+1}(1)$ be a closed minimal hypersurface with constant scalar curvature. Chern conjecture says that a closed minimal hypersurface with constant scalar curvature, $S\geq 2n$ if $S>n$, where $S$ is the squared norm of the second fundamental form. As a partial result, Peng-Terng \cite{pt1, pt2}, Yang-Cheng \cite{yc1, yc2, yc3} and Suh-Yang \cite{sy} obtained that if $S>n$, then $S>n+ \dfrac{3}n$. Recently, Chen \cite{c} has proved $S>n+\frac{26}{59}n$ if $S>n$. Since resolving Chern conjecture is a hard problem, as a midway for resolving Chern conjecture, one wants to prove $S>n+\frac12 n$ if $S>n$. In this paper, for $4\leq n\leq 13$, we give a positive solution for this midway problem. For general $n$, we prove that for a complete minimal hypersurface with constant scalar curvature, \[ S>\frac{10000000}{6869671}\,n >1.4556737869\,n>n+\dfrac{221}{485}n \] if $S>n$, which is better than the result of Chen \cite{c}.

Projection-free methods replace projections by linear minimization oracles, making their convergence fundamentally sensitive to the geometry of the feasible region. We develop a geometric framework for this dependence in Riemannian Frank–Wolfe optimization. A power-type double-geodesic modulus quantifies the intrinsic geometry of the feasible set, and an exact transfer principle shows that this modulus induces a Frank–Wolfe scaling margin modulated by the differential of the Riemannian logarithm map. The resulting factorization separates the contributions of feasible-set geometry and ambient Riemannian distortion. Rauch comparison provides explicit curvature-dependent certificates for this distortion, while on the positive-definite cone equipped with the affine-invariant metric the logarithm differential admits an exact spectral characterization in terms of relative eigenvalue spread. Combined with a quadratic error bound, the power-type geometry yields the convergence rate $O(t^{-p/(p-1)})$ for the gap-feedback short-step policy. For every fixed $p>3$, a compact feasible set in a hyperbolic plane attains the matching rate $\Theta(t^{-p/(p-1)})$ under the same policy. The geometry-derived exponent $p/(p-1)$ is therefore sharp for the gap-feedback short-step policy over the geometry-controlled class considered here.

We construct new classical global solutions of the one-phase Bernoulli free boundary problem in $\mathbb{R}^3$ whose positive phases are periodic perturbations of the product of the Hauswirth–Hélein–Pacard double hairpin domain with a line. In contrast with previous constructions of undulating cylinders, the free boundary of the base solution is not compact modulo translations. More generally, we show that the same scheme yields periodic solutions in $\mathbb{R}^{n+1}$, $n \geq 3$, bifurcating from product solutions $U(x,t)=u(x)$, when the base $u:\mathbb{R}^n\to [0,\infty)$ has nontrivial finite Morse index, is nondegenerate within a symmetry class, and satisfies certain natural geometric conditions at infinity. In dimensions $n\in\{3,4\}$, where the stable Bernstein theorem is available, the recent classification of finite Morse index solutions by Fernández-Real, Florit-Simon and Serra shows that every such solution satisfies these geometric conditions. Thus, the existence of undulating solutions in $\mathbb{R}^{4}$ and $\mathbb{R}^{5}$ bifurcating from their respective products reduces to a nondegeneracy property of the base solution. In order to confirm that the perturbed solutions are positive, we prove that finite Morse index solutions with noncompact free boundary satisfy a strong distance nondegeneracy estimate.

We prove that the Calabi flow extends past any finite time on compact Kähler surfaces if its Sobolev constants remain uniformly bounded. Moreover, we show the long-time existence of Calabi flow in certain Kähler classes on compact Kähler surfaces when the initial Calabi energy is strictly below an explicit threshold.

We study tangent flows of noncollapsed Ricci flow limit spaces in the sense of Fang-Li. We prove that the regular part of every negative time slice is geodesically convex: any minimizing geodesic that meets the regular part has its entire interior contained in it. We also identify each negative time slice with the metric completion of its regular Riemannian part, thereby identifying the negative-time tangent flow with its associated metric soliton.

We first define an extended Mabuchi functional for a class of generalized Kähler metrics of symplectic type on a toric Fano manifold $(M,J,\mathbb{T})$, and prove it is proper. Then by showing the monotonicity of functional along the normalized generalized Kähler-Ricci flow for any initial $\mathbb{T}$-invariant generalized Kähler metric with its symplectic form in $2\pi c_1(M,J)$, we prove that the flow converges to a Kähler-Ricci soliton. The result confirms a conjecture of Apostolov-Streets-Ustinovskiy.

We prove Schwarz lemmas for \(\bar\partial_b\)-harmonic maps from complete pseudo-Hermitian manifolds into K"ahler manifolds of negative curvature. For maps of bounded generalized dilatation we bound the full and the horizontal differential. For CR maps a coupled Bochner identity yields rank-dependent horizontal estimates and a bound for the Reeb differential. On Sasakian sources and complex hyperbolic targets we classify the maps attaining the horizontal bound, bound the deviation from equality in integral form by the deficit at one point, and prove compactness of almost extremal sequences. As an application, a Siu-type divergence identity yields foliation, pluriharmonicity and rank rigidity under volume growth conditions.

For K-unstable toric varieties, Székelyhidi's optimal test-function $\Theta_{P}$ is a mysterious concave function over the moment polytope $P$, which gives the maximal destabilizer for K-stability, and encodes the limiting behavior of the divergent Calabi flow. As balanced norms quantize cscK metrics, we show that $\Theta_{P}$ can be quantized by the maximal destabilizers for Chow-stability, which are given by the shortest GKZ vectors, i.e., the least norm point on the secondary polytope for the set $P\cap k^{-1}\mathbb{Z}^{n}$. Properties and algorithm for general sGKZ vectors are given. Our result may provide a new method to detect the K-unstability of toric varieties.

We construct smooth collapsing sequences with sectional curvature at least $-1$ whose limits are topological manifolds with nonzero Kirby–Siebenmann class. For every $n\geq6$, the source and limit are simply connected, have dimensions $n+3$ and $n$, and the metrics have a common positive Ricci lower bound. The limits are homeomorphic to the product of the Chern four-manifold with $S^{n-4}$. We also obtain spin examples and five-dimensional lens-space homotopy types, the latter from finite quotients of one fixed $2$-connected eight-manifold. Finally, we realize aspherical manifolds with no closed PL representatives of their homotopy types as collapsing limits with scalar curvatures tending uniformly to $+\infty$, and establish an obstruction to their approximation with almost nonnegative Ricci curvature.

The classical measure concentration phenomenon on the standard sphere $\mathbb{S}^n$ asserts that the volume of any tubular neighborhood of a totally geodesic equator $\mathbb{S}^{n-1}$ concentrates almost the entire volume of $\mathbb{S}^n$ as the dimension $n$ grows to infinity. In this paper, we extend this \lq\lq Fat Equator Effect” to arbitrary closed minimal hypersurfaces $\Sigma^{n-1} \subset \mathbb{S}^n$. By combining the analytic framework of Brownian motion and its Mean Exit Time (MET) function with the geometric tool of Schwarz symmetrization, we establish that minimality alone is sufficient to guarantee measure concentration around $\Sigma^{n-1}$ in high dimensions.

Let $(M^n,g)$ be a compact Riemannian manifold of dimension $n\ge 2$, possibly with nonempty convex $C^2$ boundary. We study the first nonzero eigenvalue $\lambda_1(M)$ of the Laplace–Beltrami operator, subject to the Neumann boundary condition when $\partial M\neq\emptyset$. Given a reference scalar curvature $R_E>0$, set $K_E:=R_E/(n(n-1))$, and define the traceless Ricci tensor $\mathring{\Ric}:=\Ric-\frac{R}{n}g$ and the Einstein deviation \[ \Theta:=|\mathring{\Ric}|+\frac{|R-R_E|}{n},\qquad \kappa_{\mathrm{Ein}}(p):=\left(\frac{1}{\vol(M)}\int_M\Theta^p\,dV\right)^{1/p}, \quad p>n/2. \] We say that $M$ is $\varepsilon$-nearly Einstein if $\kappa_{\mathrm{Ein}}(p)<\varepsilon$. The main result states that for every $\alpha\in(0,1)$ there exists $\varepsilon_0=\varepsilon_0(n,p,\alpha,D,R_E)>0$ such that if $\diam(M)\le D$ and $\kappa_{\mathrm{Ein}}(p)<\varepsilon_0$, then \[ \lambda_1(M)\ge \alpha\,\lambda_1(n,K_E,D), \] where $\lambda_1(n,K,D)$ is the first nonzero eigenvalue of the Bakry–Qian one-dimensional model problem with curvature parameter $K$, with the convention $\lambda_1(n,K,D)=nK$ when $K>0$ and $D\ge\pi/\sqrt{K}$. The proof combines the Bochner formula for the decomposition $\Ric=(n-1)K_Eg+\theta g+\mathring{\Ric}$ with an auxiliary function $J$ satisfying \[ \Delta J-\tau\frac{|\nabla J|^2}{J}-2J\Theta=-\sigma J \] and Neumann boundary conditions, together with Moser iteration and a maximum principle. As $\kappa_{\mathrm{Ein}}(p)\to0$, the estimate recovers the classical Bakry–Qian theorem.

This is the second paper in our series of works to develop the min-max theory for free boundary minimal hypersurfaces in Riemannian manifolds with boundary via the Allen-Cahn equation. Building on our previous work for general critical points, we study the boundary behaviour of limit-interface integral varifolds arising from stable (or finite Morse index) solutions to the Allen-Cahn equation with Neumann boundary condition. More precisely, we prove that the limit-interface is a curvature varifold with boundary in the sense of Mantegazza satisfying a generalized stability inequality. Furthermore, we rule out classical tangent cones at the boundary and show that the first variation of the limit-interface varifold is singular with respect to its weight measure. In particular, our result confirms a folklore conjecture that the boundary concentration examples of Malchiodi, Ni and Wei have unbounded Morse index. As will be evident in our forthcoming work, the results obtained in this paper are crucial ingredients towards the optimal boundary regularity in the Allen-Cahn free boundary min-max theory.

We prove that smooth immersed Dupin submanifolds of Euclidean space are locally Nash in arbitrary codimension and that every compact embedded Dupin submanifold is taut. These conclusions follow from our hypersurface results by the tube construction. For hypersurfaces, we require the Dupin condition only where the principal multiplicities are locally constant. No local finiteness assumption on this regular locus, or regularity assumption on its complement, is required. The proof combines a dimension-dependent degree bound on proper Dupin pieces with a continuation argument using finite jets and resultants. An analytic Hessian criterion then implies that all squared distance functions are Morse-Bott, and hence that every compact embedded hypersurface in this class is taut. In each dimension there are only finitely many such hypersurfaces up to Nash isotopy through embedded Dupin hypersurfaces. We also obtain finiteness results for taut submanifolds in arbitrary codimension and derive topological restrictions from classical classification theorems for taut embeddings.

We give a first-order Frobenius description of conjugate minimal immersions in the round three-sphere and use it to define the conjugate of an arbitrary Jacobi function. The construction is intrinsic at the linearized level and does not require the Jacobi function to be integrable through a normal deformation by minimal surfaces. We then describe how the boundary conditions are exchanged under conjugation. Along a great-circle boundary arc, a Dirichlet Jacobi function has, after a suitable ambient gauge, a conjugate Jacobi function with vanishing Neumann data. Conversely, along a reflection curve, vanishing Neumann data yield vanishing Dirichlet data for the conjugate modulo a Jacobi–Killing function. We also construct conjugate Jacobi functions for conjugate CMC~$1$ immersions in $\mathbb{R}^3$ and exhibit cylindrical counterexamples to both pure boundary exchanges, even modulo ambient Killing fields.

Let $V$ be an oriented Euclidean vector space and let $\D(V)=\Lambda V^*\otimes\Lambda V^*$ be the algebra of double forms. The two exterior factors carry natural factorwise exterior and interior multiplication operators and therefore two commuting Clifford actions. The central result of the paper is an intrinsic realization of these Clifford actions inside Greub's composition algebra. For $a\in V^*$ we introduce the inhomogeneous double form \[ \chi_a=((1\otimes a)-(a\otimes1))e^g, \] prove that the $\chi_a$ satisfy the Clifford relations for the composition product, and establish the exact regular-action formulas \[ C_R(a)\omega=\chi_a\circ\omega, \qquad C_L(a)\omega=-\omega\circ\chi_a. \] For $h\in\D^{1,1}$ this yields a canonical four-component decomposition of the coupled left–right Clifford action into exterior multiplication, double contraction and two generalized Bianchi operators. For the metric double form $g$ these four operators generate two commuting $\mathfrak{sl}_2$ actions. The construction extends canonically to arbitrary bidegree. In bidegree $(2,2)$ its degree-preserving component admits a direct comparison with the double-form extension of the sharp product: for every $R\in\D^{2,2}$, \[ \Gamma_{1,1}(R)(\omega)=\frac14R\#\omega. \] The point of this identity is not a new definition of $R\#$, which is already constructed from Clifford commutators, but the fact that it appears canonically as the central homogeneous component of the higher left–right Clifford transform. As an application, the algebraic quadratic map $\mathcal Q(R)=-\frac14R\#R$ satisfies \[ \mathcal Q(R)=-\Gamma_{1,1}(R)R, \] and its linearization at $R$ is the operator $-2\Gamma_{1,1}(R)$. For curvature tensors this is the sharp contribution to the Ricci-flow reaction term.

We prove that every biharmonic hypersurface in a space form of nonpositive sectional curvature is minimal in arbitrary dimension. This settles the hypersurface case of the generalized Chen's conjecture in hyperbolic space and gives a unified treatment of nonpositive space forms. For biharmonic hypersurfaces in the unit sphere, we derive quantitative restrictions on every possible nonconstant-mean-curvature solution. In particular, we obtain pointwise criteria forcing constant mean curvature, a strict scalar-curvature bound in dimensions at least five, and local classification results above the classical CMC gap threshold. These results provide further evidence for the BMO conjecture.

We develop a theory of orientability for metric manifolds and singular metric spaces using locally integral currents. For topological $n$-manifolds with locally finite Hausdorff measure, we introduce local degree and measure-theoretic conditions under which topological orientations correspond canonically to boundaryless, locally integer rectifiable $n$-currents of multiplicity one on the rectifiable part. We extend these currents to a class of singular spaces whose manifold part is dense and of full measure, and establish constancy and top-dimensional homology results under local uniqueness of orientation. For orientable non-collapsed $\RCD(\kappa,n)$ spaces, we prove a current-theoretic Stokes theorem: the boundary of the orientation current is the induced orientation current of the geometric boundary, with multiplicity one. In particular, its mass measure is precisely the $(n-1)$-dimensional Hausdorff measure on that boundary. For purely $n$-dimensional, locally geodesically complete spaces with local upper curvature bounds, we show that local uniqueness of orientation excludes codimension-one branching and implies a local $1$-Poincaré inequality. The latter result applies beyond homology manifolds and extends the connection between quantitative topology and Poincaré inequalities to this class of singular spaces.

In this paper we introduce a geometrically constrained optimal transport problem on Riemannian manifolds endowed with a Borel foliation, in which mass is transported only within individual leaves. We characterize the existence of foliated transport plans through the equality of the transverse projections of the marginals and show that the corresponding quadratic transport cost decomposes into a measurable family of leafwise optimal transport problems. We also formulate the problem as a classical Kantorovich problem with an extended-valued leafwise cost. Finally, we introduce a foliated continuity equation and establish a Benamou-Brenier formula relating the static and dynamical foliated formulations.

The hydrodynamical Hopf-Rinow problem asks whether a given reconfiguration of an ideal fluid can be attained by an Euler flow. On the flat two-dimensional torus, we give an equivalent criterion for the strong version of this problem (which requires that the fluid motion minimizes the total kinetic energy required) in terms of metric betweenness. Toward this, we prove that sufficiently small normal neighborhoods in the volume-preserving diffeomorphism group of the flat $n$-dimensional torus, $n \geq 2$, equipped with the kinetic energy metric, are geodesically convex; resolving the strong problem locally in the $H^s$ topology. We further establish local bi-Lipschitz equivalence of the intrinsic geodesic distance and the extrinsic $L^2$ distance. Finally, we prove that the squared geodesic distance from the identity admits a $C^{1,1}$ extension from a sufficiently small normal neighborhood to the ambient $L^2$ space.

We prove the sharp horizontal isoperimetric inequality in $\mathbb H^n$, $n\ge2$, with equality precisely for left translates and Heisenberg dilations of Pansu balls, up to Haar-null sets.

For $n\ge2$, we prove a quantitative centre-rigidity estimate for bounded sets of finite Riemannian perimeter that are stationary and volume-constrained stable for normalized Riemannian perimeter with prescribed curvature $h_j$. Under confinement to a fixed compact set and cutoffs of vanishing full surface Dirichlet energy on the smooth regular boundary, the squared $L^2$ norms of the horizontal centre derivative, angle defect and weighted shape defect are bounded by $C(\varepsilon_j^2+\|h_j-2n\|_{C^2})|\Omega_j|$ when the parenthesized quantity is sufficiently small. The excess of normalized Riemannian perimeter over horizontal perimeter satisfies the same bound.

For $n\ge2$, we prove that the $C^2$ boundary of a bounded nonempty open set in $\mathbb H^n$ is volume-preserving stable if and only if it is a Pansu sphere up to left translation and Heisenberg dilation. The proof combines centre reconstruction with a volume-constrained test field derived from Pansu dilation. Consequently, every isoperimetric region admitting an open representative with $C^2$ boundary is a Pansu ball up to these symmetries.

For $n\ge2$, we establish second differentiability of horizontal perimeter for compact $C^2$ boundaries in $\mathbb H^n$ and the continuous extension of their affine second variation to $C^1$ vector fields along the boundary. For the unit Pansu sphere, the Hessian agrees with the closed Jacobi form across both characteristic poles, including variations which move them. We compute the full Friedrichs spectrum of its Jacobi operator. The operator has compact resolvent, Morse index one and nullity $2n+1$; its kernel consists exactly of normalized translation speeds. Consequently, the sphere is stable under volume-preserving ambient variations.