Papers

We study the directed completion of Lorentzian pre-length spaces, and we show that, under some natural assumptions which cover the case of smooth globally hyperbolic spacetimes, it coincides with the future causal completion of Geroch–Kronheimer–Penrose. Moreover, we provide applications of our findings by characterizing the directed completion of the Kruskal–Szekeres spacetime in terms of its radial null geodesics.

We study the operator symbols of Dirichlet type introduced by Hedenmalm and Shimorin (2020), in connection with a given contraction on $L^2$ of the unit disk. They are always holomorphic functions on the bidisk. Such Dirichlet symbols associated with the Grunsky operator of a univalent function on the disk or exterior disk are of particular significance. From the work of Hedenmalm and Shimorin, we know they are characterized as solutions of a certain nonlinear wave equation. We perform a local analysis of such symbols near the diagonal on the bidisk, and in so doing, we provide alternative chart coordinates for the infinite-dimensional manifolds of univalent functions of the (exterior) disk. Those coordinates allow us to characterize $\log\psi'$ for $\psi$ in the class $\Sigma$ of normalized univalent functions without explicitly touching the univalence property. Moreover, that manifold extends the universal Teichmüller space of Lipman Bers beyond the quasicircle boundary setting, allowing for even more fractality. The fractality of harmonic measure for the domain associated with the given univalent function can be studied in terms of the asymptotic variance introduced by McMullen (2008). The asymptotic variance captures the $L^2$ average amplitude of the nonlinearity. We here introduce the new concept of Schwarzian asymptotic variance, which measures the average amplitude of the Schwarzian derivative in place of the nonlinearity. For this new Schwarzian asymptotic variance, we find that the effective average amplitude of $(1-|z|^2)^4|\Sop(\vp)|^2$ on the disk in the hyperbolic metric sense is at most $9.07735\ldots$, considerably smaller than the maximum amplitude of $36$. Here, $\Sop(\vp)$ is the Schwarzian derivative of $\varphi\in\mathscr{S}$, and the analogous statement is valid for $\psi\in\Sigma$ as well.

Let $\mathcal{H}_d(M)$ denote the space of harmonic functions with polynomial growth of degree at most $d$ on a complete Riemannian manifold $(M,g)$. Yau raised two fundamental questions regarding $\mathcal{H}_d(M)$ on complete manifolds with nonnegative Ricci curvature. The first question is the finite dimensionality of $\mathcal{H}_d(M)$, which was confirmed by Colding and Minicozzi. The second question asks whether a sharp upper bound given by its Euclidean analog $\operatorname{dim}\mathcal{H}_{d}(\mathbb{R}^n)$ holds. We verify that the second question is true on locally conformally flat manifolds. Indeed, one can precisely determine the value of $\dim \mathcal{H}_d(M)$ case by case.

Classical definitions of envelopes are vague for singular curves in the Lorentz-Minkowski plane. To study envelopes of such curves, we introduce a family of smooth curves that may admit singular points. Notably, the curves considered here may contain both non-lightlike and lightlike points. We then propose a new definition of envelopes for such curves and investigate their properties using lightlike tangential data. Furthermore, we examine the contact between the curves and their envelopes at singular points.

The Calabi-Yau conjectures for complete minimal hypersurfaces $\Sigma^{n}\subset \mathbb{R}^{n+1}$ for $n\geq 2$ ask whether a complete minimal hypersurface must be unbounded, and more strongly whether it must be proper. In this work, we resolve this conjecture for complete, connected, embedded minimal hypersurfaces $ \Sigma^3 \subset \mathbb{R}^4$ with bounded second fundamental form and finite second Betti number $b_2(\Sigma;\mathbb{Z}_2)<\infty$.

We construct a $4$-parameter family of properly embedded genus-zero surfaces with at most two boundary components in a Riemannian $3$-ball. The construction is a free boundary analog of the canonical $5$-parameter family of surfaces on the $3$-sphere by Marques-Neves in the proof of Willmore conjecture. In the Euclidean unit ball, this family realizes the critical catenoid as a Simon-Smith min-max limit. As applications, we give an upper bound of Almgren-Pitts $4$-width $\omega_4(\mathbb{B}^3)$ by the area of the critical catenoid. Moreover, we prove the existence of at least three free boundary minimal annuli in any compact $3$-manifold with nonnegative Ricci curvature and strictly convex boundary.

We prove several analytic results on (possibly noncompact) complete singular surfaces having locally bounded integral curvature (in short: BIC surfaces). Regarding these as metric measure spaces with the 2-dimensional Hausdorff measure, we show that these are infinitesimally Hilbertian, locally doubling and satisfy a local Poincaré inequality. In particular, this entails the existence of a jointly Hölder continuous heat kernel for the Cheeger Laplacian. Assuming that the negative part of the curvature measure of a BIC surface satisfies a Dynkin-type condition, we show that the surface is bi-Lipschitz equivalent to a BIC surface with a lower bounded curvature measure, entailing global variants of the aforementioned results.

In this paper, we study the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$. This problem was recently posed by Sadovsky and Zhang. First, by a second variation argument and a dimension reduction construction, we show that the inequality fails for arbitrary convex bodies when $q>n$, and fails even in the origin-symmetric class when $q>n+2$. Secondly, we prove the endpoint case $q=n+2$ for origin-symmetric convex bodies via Hadwiger's inequality for the polar moment of inertia. Finally, for unconditional convex bodies, we establish the inequality in the full range $0<q\le n+1$ by using a singular weighted Reilly formula and a coordinate-slice Hardy inequality. As applications, we derive several uniqueness results for the corresponding dual curvature measures.

The method of complex-valued eigenfamilies has been used to solve difficult non-linear problems in differential geometry. The purpose of this note is to generalise the construction method from the use of polynomial solutions to those obtained by rational exponents. This gives many new solutions to the problem under investigation.

We prove a Nakai-Moishezon-type criterion for complex Hessian-type equations on projective manifolds whose associated degree-$n$ polynomials are strongly strictly right-Noetherian. For strictly right-Noetherian polynomials of arbitrary degree, we prove a uniform Nakai-Moishezon-type criterion. This class includes the complex Hessian and Hessian quotient equations.

The cone $M \times \mathbb{R}_+$ over a Sasakian manifold $M$ is equipped with a canonical Kähler structure with specific homogeneity properties with respect to the $\mathbb{R}_+$ coordinate. This Kähler structure completely encodes the underlying Sasakian structure. Recently, Grabowski, Grabowska and Mohseni provided a broader conceptual framework for this phenomenon via homogeneous Kähler structures, i.e. Kähler structures on a principal $\mathbb{R}^\times$-bundle $P$ satisfying similar homogeneity properties. This approach successfully extends Sasakian geometry from cooriented contact manifolds (where $P$ is a trivial principal bundle) to non-necessarily coorientable contact structures (where $P$ is non-necessarily trivial). Homogeneous Kähler structures are genuinely more general than Sasakian structures and this note precisely characterizes the extent of this generalization. This is achieved through a detailed analysis of all the involved compatibilities in terms of the line bundle tautologically associated to $P$. We also show that modifying the homogeneity condition on the Kähler structure allows this framework to encompass co-Kähler structures and a natural generalization of those as well.

We develop a unified geometric and probabilistic theory of conformal type for unimodular random infinite trivalent hyperbolic polyhedra in $\mathbb{H}^3$. By corresponding these with dual angled disk triangulations and regular circle patterns, we associate to each face an intrinsic geometric characteristic number $L_f(P)$, determined entirely by local dihedral geometry. For the root face $f$, we establish the unimodular Gauss-Bonnet formula $\mathbb{E}[L_f(P)] = 2\pi - (\pi/3)\mathbb{E}[deg(f)]$. Under natural tameness and admissibility assumptions, this yields a sharp dichotomy: a unimodular random trivalent hyperbolic polyhedron is parabolic precisely when $\mathbb{E}[L_f(P)] = 0$, and hyperbolic when $\mathbb{E}[L_f(P)] < 0$. Thus, global conformal type is governed by the expectation of a local geometric quantity. We also investigate the approximation of infinite polyhedra by finite ones. We prove that every admissible Benjamini-Schramm limit of uniformly face-rooted finite trivalent hyperbolic polyhedra is necessarily parabolic, revealing a geometric and topological obstruction to the existence of hyperbolic unimodular polyhedral limits. To study stochastic behavior in the hyperbolic regime, we overcome the failure of classical circle packing tools for unbounded degrees by establishing a refined ring lemma for regular circle patterns. This yields effective exponential control of adjacent circle radii via local flower degrees. Combined with boundary methods, we identify the Poisson boundary with the circle at infinity and prove positive hyperbolic speed for the face random walk. These results provide the first quantitative framework connecting local three-dimensional dihedral geometry, global conformal type, and asymptotic stochastic behavior of unimodular random infinite hyperbolic polyhedra.

We prove that every complex projective variety is birational to a smooth projective manifold admitting a constant scalar curvature Kähler (cscK) metric. For any smooth projective variety, a birational cscK model is obtained by resolving the codimension two base locus of a general Lefschetz pencil in a sufficiently positive linear system. The cscK polarization is given explicitly, producing a cscK model also when the initial variety is unstable. In dimension two this proves the folklore conjecture that the blowup of any complex projective surface in enough points admits cscK metrics.

We construct two families of curvature inhomogeneous Riemannian manifolds with constant Ricci eigenvalues. The first, derived from the Einstein warped products, has two distinct Ricci eigenvalues and admits a local isometric immersion of minimum codimension two. The second, arising from the Riemannian Schwarzschild–Tangherlini manifold, has $k+1$ distinct Ricci eigenvalues and admits an isometric embedding of codimension $k+2$, which is the smallest within the adapted product class.

Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $\omega=i\partial\overline{\partial}\phi$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\,\omega\leq\omega$ and the shortest nonconstant closed geodesic has length at least $2\pi$, then \[ K_{m\phi}\geq \frac{2m-1}{4\pi}\,\omega, \] with sharpness holding for $(\mathbb P^1,\mathcal O_{\mathbb P^1}(2))$. We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound $-\omega\leq\mathrm{Ric}\,\omega\leq\omega$ and the same closed-geodesic hypothesis, we also prove \[ K_{m\phi}\leq \frac{m\omega}{2\pi} \left(1+\frac{3}{2m}\right). \] The lower estimates use the deformation to the tangent space version of the Ohsawa–Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki–Zwonek and isoperimetric inequalities.

We prove positive-time regularity for harmonic map heat flows from finite-dimensional $\mathrm{RCD}(K,N)$ spaces into complete $\mathrm{CAT}(0)$ spaces, without assuming that either the source or the target is smooth. For bounded-image initial data, the $\mathrm{EVI}$ gradient flow of the Dirichlet energy admits a representative that is locally Lipschitz jointly in space and time. Moreover, its spatial pointwise Lipschitz constant satisfies an Eells–Sampson-type parabolic Bochner inequality. The main difficulty is that the smooth parabolic perturbations used to select contact points are unavailable on an $\mathrm{RCD}$ source. We overcome it by a sliced contact-selection principle that converts the elliptic ABP estimate on $\mathrm{RCD}$ spaces into the space–time contact selection required by the Hamilton–Jacobi argument.

We establish first and second variation formulas for general mean curvature functionals under arbitrary variations in arbitrary Riemannian manifolds. We obtain corresponding formulas in the sub-Riemannian Heisenberg group through a Riemannian approximation scheme.

We introduce a weighted version of intermediate Ricci curvature and establish several equivalent characterizations of its lower bound. As an application, we generalize the results of Aishwarya–Rotem–Shenfeld [arXiv:2509.23399v1] by deriving intrinsic-dimensional evolution variational inequalities and the corresponding Wasserstein contraction estimates for the heat flow. We also compare our characterization with that of Ketterer–Mondino [arXiv:1610.03339v3] via lower-dimensional optimal transport.

We construct new families of complete minimal submanifolds of codimension two in the non-compact duals to the classical compact Lie groups $\SOC n/\SO n$, $\SLC n/\SU n$ and $\SpC n/\Sp n$.

A Heisenberg Bieberbach manifold is a compact quotient of the Heisenberg group by a discrete torsion-free subgroup of its pseudo-Hermitian automorphism group. In this paper, we study the joint spectrum of the sub-Laplacian and the Reeb vector field on three-dimensional Heisenberg Bieberbach manifolds. In particular, we explicitly compute the multiplicities of the joint spectrum using theta functions and the character theory of finite groups.

Under an explicit formal mapping-space hypothesis, we develop a facewise formulation of the classical AKSZ construction on compact oriented manifolds with ordinary corners. The codimension-$r$ data—a mapping space carrying a closed two-form of degree $r-1$, an action of degree $r$, and a cohomological vector field—and the modified Batalin–Vilkovisky/Batalin–Fradkin–Vilkovisky Hamiltonian identity relating consecutive strata are those of the maximally extended BV–BFV theory of Cattaneo–Mnev–Reshetikhin. What is added here is the organization over the entire face poset: the Hamiltonian defect on a face is the sum of the pullbacks of the primitives on its codimension-one faces, weighted by the orientation incidence numbers, so that the boundary term of the single-stratum identity is resolved into its connected pieces with signs. Organizing these defects by the face incidence complex yields a total-complex theorem: factorially normalized facewise transgression is a cochain map, so closed target forms transgress to cocycles, and the twice-iterated defect vanishes because the signed face differential squares to zero. We verify all four codimension-two cancellations explicitly for four-dimensional BF theory on $M=\Gamma\times[0,1]^2$. We also establish a reduction criterion for singular corner data. If a raw codimension-two descendant is presymplectic and its reduced Dirac structure is the graph of a Poisson bivector, the shifted cotangent construction gives a canonical strict degree-two corner theory. The passage from a reduced Poisson bivector to a strict corner theory is already recorded in \cite{CFT2026}; what is isolated here is the hypothesis under which it applies, and its relation to the face-incidence structure. The construction provides a rigorous ordinary-corner benchmark for extensions of AKSZ descent to Joyce generalized corners.

We study rational valued counts of geodesic strings (reparametrization equivalence classes of closed geodesics) for complete Riemann-Finsler manifolds, based on the Fuller index of the geodesic flow. The main conceptual result is a product type formula for these counts. Combined with aspects of KAM theory, it yields the following sample phenomenon. Let $g$ be a generic Finsler metric on $T ^{2}$, sufficiently $C ^{\infty }$-close to a flat metric, and fix a prime $p$ and a nontrivial free homotopy class $\beta $. If there is a class $\beta$ $g$-geodesic string with multiplicity divisible by $p$, then there is another one. We also obtain arithmetic constraints on counts of closed geodesics in mapping tori and flat bundles, and constraints on the existence of negative sectional curvature metrics. These counts can be understood as a shadow of a conjectural orbifold Morse homology of the infinite-dimensional quotient stack $[LX/S^1]$.

We study normalized Ricci flow on simply connected homogeneous spaces G/H for which the isotropy representation splits into exactly two equivalent irreducible subrepresentations. We prove that every G-invariant metric evolves to one with positive Ricci curvature, and that the family of G-invariant metrics with positive Ricci curvature is forward-invariant under the flow. The proof relies on the fact that the phase portrait of the family of fixed-volume G-invariant metrics can be explicitly visualized.

Let $(M,g)$ be a connected, compact, $n$-dimensional Riemannian manifold with $\operatorname{Ric}(M,g)\geq-(n-1)\kappa g$. We introduce a weighted combinatorial Laplacian on $\varepsilon$-discretizations of $M$ and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operator. More precisely, the eigenvalues of the two operators are uniformly comparable with constants depending only on $n,\kappa,\varepsilon$, independently of the injectivity radius. As an application, we prove spectral stability under measured Gromov-Hausdorff convergence. We also recover the Schoen-Wolpert-Yau inequality using the weighted discretization on families of pinching genus-$2$ hyperbolic surfaces.

We study the spiral product $G_\gamma(t,x,y)=(z_1(t)f_1(x),z_2(t)f_2(y))$, which couples two spherical $\mathscr C$-totally real immersed factors through a profile curve $\gamma=(z_1,z_2)\subset\mathbb S^3$. Its minimality is governed by a weighted-geodesic system: $G_\gamma$ is minimal precisely when both factors are minimal and $\gamma$ is an unparametrized geodesic of $|z_1|^{2k_1}|z_2|^{2k_2}g_{\mathbb S^3}$. This flow is Liouville integrable and, when $k_1+k_2>0$, its phase map has an open dense full-rank locus. Consequently, closed profiles of arbitrarily large primitive order occur densely. For compact minimal factors, Routh reduction and Sturm oscillation give $\text{Ind}(G_\gamma)\geq \text{Ind}(f_1)+\text{Ind}(f_2)+2m_\gamma-3$ for a closed oscillatory profile of primitive closing order $m_\gamma$. On the contact level, a factor-adapted choice of profiles produces, from any prescribed pair of compact connected embedded special Legendrians, embedded special Legendrian products of every sufficiently large prime closing order. Finally, canonical finite horizontal lifts and Hopf projection give Delaunay-type minimal Lagrangians in complex projective spaces. For compact embedded inputs and ordinarily closed profiles, the primitive spherical quotient is embedded, while its projective quotient is embedded exactly when the reduced relative winding is one.

We prove a new Carnot-scale $n$-step nilpotent $\pi_1$-de Rham theorem: along every convergent scale sequence, the asymptotic homotopy class of a long trajectory is identified with the nilpotent development of its macroscopic horizontal path. The key analytic mechanism is an asymptotic-development theorem showing that uniform convergence of rescaled horizontal paths, together with a uniform bound on variation, forces uniform, layer-by-layer convergence of their full Carnot-rescaled nilpotent developments. The theorem recovers Schwartzman's theory of asymptotic cycles in step one and the 2-step asymptotic homotopy theory of Benardete and Mitchell in step two, while extending the correspondence to arbitrary nilpotent step.

We prove that the Schwarz' Primitive (P) and Diamond (D) surfaces are local minimizers of the variance of Gaussian curvature among local deformations within the moduli space of triply periodic minimal surfaces of genus 3 (TPMSg3s). Our approach interprets the branch values of the Gauss map as a configuration of eight points on the sphere and expresses the variance of Gaussian curvature as the product of two integrals involving exponentials of Green's functions. We then show that the Hessian of this functional is positive definite at the cubic configuration when restricted to antipodal deformations, thereby establishing the local minimality of the P and D surfaces. Along the other deformation directions, the Hessian is positive definite except for a two-dimensional eigenspace with slightly negative eigenvalues, leaving promising hope that the Gyroid is also a local minimizer.

For solutions to the Minkowski problem of the even dual curvature measures $\widetilde{C}_q$ in $\mathbb{R}^n$, $n\ge 2$, we prove the non-uniqueness for $q>n$ and $n\ge 2$, and the uniqueness for $0<q<n$ and $n=2$. These results are governed by our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals $\widetilde V_q$.

AAn old conjecture in non-Kähler geometry states that any compact Hermitian manifold with constant Chern holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but still open in dimensions 3 or higher, except for several special classes of Hermitian manifolds. For the important class of locally conformally Kähler manifolds, the conjecture was proved by H. Chen, L. Chen, and Nie in 2021 when the constant holomorphic sectional curvature is non-positive and the remaining case was solved recently by Huang and Wan using the result of Kamishima on Bochner-Kähler manifolds. In this article, we use their technique to answer similar questions for locally conformally Kähler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature.

We construct a family of $\mathbb{R}/\mathbb{Z}$-valued CR invariants for compact strictly pseudoconvex CR manifolds admitting pseudo-Einstein contact forms. We define these invariants by applying the theory of Cheeger–Simons differential characters to a globally defined modification of the normal tractor connection. When the CR holomorphic tangent bundle is trivial, their natural $\mathbb{R}$-valued lifts agree with the generalized Burns–Epstein invariants. For CR manifolds arising as boundaries of relatively compact strictly pseudoconvex domains, we identify these differential characters with those determined by the renormalized connection and derive bulk–boundary formulas involving renormalized characteristic numbers and residues of Chern classes. These formulas recover and extend the results of Burns–Epstein and Marugame. We also obtain obstructions to CR embeddings into complex Euclidean space.