Papers

We study corank-two distributions $D$ of odd rank $2N+1$ whose associated pencil of skew-symmetric forms $\{d\alpha|_{D}(q)\mid \alpha\in\Omega^1(M),\ \alpha|_{D}=0\}$ lies, at every point, in the generic orbit of the natural $\mathrm{GL}\bigl(D(q)\bigr)$-action; equivalently, $D$ has maximal first Kronecker index. In corank two, this class is the analogue of contact and even-contact distributions. We prove that the corresponding differential relation is ample and hence satisfies the multi-parametric $C^0$-close $h$-principle, and we deduce a complete topological characterization for a $(2N+3)$-dimensional manifold to admit such a distribution. The criterion is satisfied by non-parallelizable manifolds as well. To the best of our knowledge, ampleness has been verified so far for two classes of distributions defined by an open $\mathrm{Diff}$-invariant relation: even-contact distributions (McDuff, 1987) and distributions of rank greater than two with maximal small growth vector, in arbitrary ambient dimension (Martínez-Aguinaga, 2026); for $N\ge2$, our class is a proper open subclass of the latter, singled out by a further condition on the pencil. The verification of ampleness reduces, via Kronecker–Weierstrass normal forms of the restrictions of the pencil to hyperplanes, to computing the convex hulls of the connected components of the complement of the resultant hypersurface in the space of pairs of real binary forms of fixed degrees, components separated by a winding number, i.e., by the classical Cauchy index. This is in contrast with the even-contact case of McDuff, where the set to be removed is thin (i.e., of codimension at least two), so that its complement is automatically connected and ampleness is immediate.

Let $M_f$ be the closed hyperbolic $3$–manifold which is the mapping torus of a pseudo-Anosov homeomorphism $f$ of a closed surface $S$ of genus at least two. The inclusion of the fiber subgroup \(\pi_1(S)\hookrightarrow \pi_1(M_f)\) has a Cannon–Thurston boundary map \(\partial\pi_1(S)\cong S^1\to S^2\cong\partial\pi_1(M_f)\), which is a Peano curve. We relate this map to two variational representatives of the fiber homotopy class: the harmonic map of Eells–Sampson and Hartman, and a least-area embedded minimal surface. Equivariant lifts of homotopic maps are a bounded distance apart; hence, for every marked conformal structure on $S$, the lift of the harmonic representative extends continuously to the closed disk with boundary value the Cannon–Thurston map, and a least-area fiber lifts to a properly embedded minimal plane in $\Hh^3$ with the same boundary map. We give area and curvature bounds for a least-area fiber, show that its largest principal curvature is at least one, and study the energy of the harmonic representatives as a function on Teichmüller space, its critical points, which are the zeros of the Hopf differential, and its invariance under the monodromy.

Let $g=w|dz|^2$ be an affine special Kähler metric on a punctured disc and let $\Xi=q(z)\,dz^3$ be its associated holomorphic cubic differential. We prove that if the $g$-area of concentric punctured neighbourhoods grows at most polynomially, then $q$ is meromorphic at the puncture. More precisely, an area bound of order $\eps^{-N}$ forces every pole of $q$ to have order strictly less than $N+3$. Hence an essential singularity of the cubic differential rules out every polynomial area upper bound. The proof associates to $(g,\Xi)$ a curvature $-1$ conformal pseudometric and applies the Ahlfors–Schwarz lemma, followed by the submean inequality for holomorphic functions. We also discuss the identically zero cubic differential and the borderline logarithmic growth responsible for the strict pole-order bound.

We prove that, in every closed Riemannian three-manifold $(M^3,\bar g)$, there exists a constant $C>0$ such that every closed smoothly embedded minimal surface $\Sigma\subset M$ satisfies $b_1(\Sigma;\mathbb{Z}_2)\le C\bigl(\operatorname{Ind}(\Sigma)+\operatorname{Area}(\Sigma)\bigr)$. The estimate holds without orientability or two-sidedness assumptions and establishes the additive genus-index-area estimate conjectured by Song. For closed connected orientable two-sided minimal immersions, we also obtain explicit genus bounds under a lower bound on the ambient sectional curvature. Under positive ambient Ricci curvature, we prove the universal inequality $\gamma(\Sigma)\le 8\operatorname{Ind}(\Sigma)$.

We develop a general construction of homogeneous solutions to the Bernoulli free boundary problem, as well as general extremal domains on the sphere, from isoparametric foliations of the sphere. Our construction produces rich families of infinitely many new examples with sophisticated topologies connected to minimal surfaces of the sphere by smooth families of interpolating capillary surfaces, which include novel singularity models in low dimensions and recover most known homogeneous one-phase solutions. We also introduce a geometric representation for every such homogeneous map and establish a geometric rigidity theorem: for every prescribed isoparametric foliation, and in particular for cohomogeneity-one subgroups of $O(n)$, its radial geometry uniquely determines the corresponding homogeneous solution. Finally, we study the density of the constructed solutions and conjecture that the lowest density among non-flat cones in a given dimension $n \geq 5$ is attained by an $O(k) \times O(n-k)$-invariant cone.

We verify that the stable minimal hypersurfaces with prescribed singular sets constructed by Leon Simon are area-minimizing. Thus, Simon's work settles the existence of area-minimizing hypersurfaces with fractal singular sets.

For a convex mean curvature flow in $\mathbb{R}^{n+1}$, the corresponding arrival time function solves a degenerate elliptic equation that becomes singular at extinction. We prove that it is $C^{24,\alpha}$ for any $0<\alpha<1$ in the plane, with a logarithmic modulus for its twenty-fourth derivatives. We prove that this bound is sharp combining regularity obstruction in our companion paper. We obtain asymptotic expansions of the rescaled flow to any prescribed order, including derivative estimates for the remainder. Combined with the work of Sesum for $n\ge 2$, this also gives the sharp $C^{2,2/n}$ regularity for convex arrival time functions in higher dimensions.

We study the regularity of the arrival time function associated with mean curvature flow, formulated as a degenerate elliptic equation encoding the singular structure of the flow. We identify a systematic mechanism obstructing higher regularity, arising from higher-order asymptotic expansions near spherical singularities. We construct uncountably many low-regularity arrival time functions in all dimensions. In particular, we settle the question of smoothness in the planar case by showing that even for convex curve shortening flow the arrival time need not be $C^{25}$. Our companion paper proves that every such arrival time is $C^{24,\alpha}$ for any $0<\alpha<1$. Hence the obstruction at order $25$ gives the optimal regularity threshold. Our approach introduces new analytic ingredients, including a precise correspondence between elliptic asymptotic expansions and parabolic long-time asymptotics of the associated rescaled mean curvature flow, a complexification of the arrival time equation, and a method for prescribing higher-order asymptotic expansions.

Let $n \geq 4$ and $\rho: \pi_1S \rightarrow PSL(n,\mathbb{R})$ be a Hitchin representation. We study the topology of a cocompact domain of discontinuity $\Omega_{\rho}$ in the flag manifold $\mathcal{F}_{1,n-1}$ of line-hyperplane pairs in $\mathbb{R}^n$ defined by Guichard-Wienhard. In particular, we lift $\Omega$ to its $(\mathbb{Z}_2\times \mathbb{Z}_2)$-cover $\hat{\Omega}$ in the Stiefel manifold $V_2(\mathbb{R}^n)$. The domain $\hat{\Omega}$ is highly connected and smoothly fibers over hyperbolic space $\mathbb{H}^2$ with unknown fiber $\hat{\mathfrak{F}}$, a closed $(2n-5)$-manifold. We determine the homeomorphism type of $\hat{\mathfrak{F}}$ as well as its diffeomorphism type up to connected sum with a homotopy sphere. The classification of $\hat{\mathfrak{F}}$ involves computing both its homology and a certain differential topology invariant needed to invoke Wall's classification of highly connected odd-dimensional manifolds.

For an $n$-Lie–Rinehart algebra, $n>2$, the anchor is defined on $\Aext{n-1}L$, so its kernel does not determine a distinguished submodule of $L$. We introduce the anchor radical $\Rad(\rho)\subset L$ and prove that it is an adjoint-stable ideal on which the bracket is $A$-multilinear. We then derive an explicit obstruction to the $A$-linearity of metric compatibility in fundamental directions. This obstruction vanishes on $\Rad(\rho)$, whereas a strongly nondegenerate compatible metric on all of $L$ forces the anchor to vanish. These results lead to a natural notion of orthogonal structure on the anchor radical. For anchors induced by Nambu tensors, the radical is identified with the cotangent annihilator of the tensor.

Bi-invariant geodesic regression generalizes linear regression to Lie groups. Its main feature is that it respects the symmetries of the group so that the resulting estimator is independent of arbitrary choices such as a reference frame. However, the local existence and uniqueness of the underlying estimator have not been shown until now. Furthermore, the convergence properties of the proposed algorithm for computing the estimator are not known. In this work, we investigate these questions. We prove that, locally, a unique estimator exists and give explicit bounds on the size of this neighborhood. We also show that the proposed iterative algorithm converges linearly to this estimator.

We develop several tools to study the local and global structure of the Teichmüller and moduli spaces for complex nilmanifolds. As a starting point, we prove a structure theorem for holomorphic maps between complex nilmanifolds, which allows for an explicit description of the Teichmüller and moduli spaces. We then introduce two period maps, and give general criteria for the existence of a complex structure on these spaces. To illustrate the theory, we prove a global Torelli theorem for various classes of nilmanifolds, including principal torus bundles over tori and almost abelian nilmanifolds. In particular, we show that there are nilmanifolds of arbitrary dimension and nilpotency index for which the Teichmüller space admits the structure of a complex manifold.

We introduce a new method for constructing Zoll metrics that admit first integrals polynomial in the momenta. The method uses Funk's form of the metric together with an algebraic relation between the first integrals. This reduces the problem to an algebraic system from which all unknown functions can be determined explicitly. In the case of a non-trivial cubic integral, we present an explicit two-parameter family of metrics that covers all such rotationally symmetric Zoll metrics on the $2$-sphere: every such metric is isometric to a member of this family. We compare these metrics with the earlier results of Valent, Duval, and Shevchishin.

In this paper, we apply Nash-Moser iteration and Saloff-Coste's Sobolev inequalities to derive gradient estimates for positive solutions to a class of nonlinear parabolic equations of the form \begin{equation*} u_{t}=\Delta u^{m}+a(x, t)u\log u+b(x, t)u, \quad 1-\frac{2}{n}<m<1 \quad \text{or}\quad 1<m<1+\frac{1}{1+\sqrt{2n}}, \end{equation*} on a complete Riemannian manifold $(M,g)$ of dimension $n$, where $a(x, t)$, $b(x, t)$ are $C^{1}$ functions. For $a=b=0$, this equation becomes the porous medium equation (PME) or the fast diffusion equation (FDE) when $m>1$ or $m<1$, respectively. As consequences, we obtain Liouville type theorems for the corresponding constant coefficient equations when \begin{equation*} 1-\frac{1}{\sqrt{n-1}}<m<1\quad \text{and}\quad 1<m<1+\frac{1}{\sqrt{n-1}}. \end{equation*}

We establish local Hamilton-type gradient estimates for positive $C^{2,1}$ solutions of $u_t=\Delta u^m$ on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed $m>1$ and every fixed $0<m<1$, there exist $\beta=\beta(m,n)>0$ and $C=C(m,n)>0$ such that a solution $0<u\leqslant A$ in $B_{2R}(x_0)\times(t_0-T,t_0]$ satisfies the following local gradient estimate \begin{equation*} \sup_{B_R(x_0)\times(t_0-T/2,t_0]}|\nabla u^\beta| \leqslant C A^\beta \left(\frac1R+\sqrt{k}+\frac{A^{(1-m)/2}}{\sqrt T}\right), \end{equation*} where $\mathrm{Ric}_M\geqslant-k$. The proof uses an intrinsic quantitative alternative to locate, around each prescribed positive point, a cylinder on which the solution has a controlled upper-to-lower ratio. A local gradient estimate on this cylinder is combined with a stopping argument. As a consequence, every uniformly bounded positive ancient solution on a connected complete manifold with nonnegative Ricci curvature is constant.

We study the Wasserstein lift of nonlinear Poisson systems with inhomogeneous Neumann boundary conditions. We establish the existence of minimizers using new estimates involving the $p$-moments of measure-valued traces. We also derive the optimality conditions, which turn out to be matrix-valued generalizations of pressureless Euler equations. While deterministic minimizers exist under suitable convexity assumptions and for one-dimensional source domains, we show that genuinely measure-valued minimizers arise in higher dimensions. In particular, we construct examples in which the lifted and classical variational problems have different minimum values, thereby precluding the solutions to the lifted problem from being deterministic, despite the absence of any a priori mechanism enforcing measure-valued behavior. To the best of our knowledge, this phenomenon is new in the nonlinear elliptic theory.

We prove that at any nondegenerate neck pinch singularity of Lagrangian mean curvature flow of surfaces the mean curvature becomes unbounded, with control on the blow-up rate.

We find bounds for the genus and for the number of boundary components of a surface that is either index one as a free boundary minimal surface or stable as a free boundary constant mean curvature surface in a Riemannian three-manifold with convex boundary and non-negative Ricci curvature, provided the first Dirichlet eigenvalue of its Jacobi operator is non-negative. As an application, we establish strong topological restrictions on such surfaces in compact strictly convex domains of three-dimensional normal homogeneous spaces.

We prove the (uniform) Yau-Tian-Donaldson Conjecture for transcendental Kähler classes in the case of trivial automorphisms. Indeed, we show the existence of Special Kähler Fujita Approximations of big cohomology classes associated to big test configurations, establishing the equivalence between the uniform $K$-stability and the strengthened version for models. More generally, we show that any big cohomology class on a compact Kähler manifold admits such Special Kähler Fujita Approximations: the volume and the analogue of the Riemann-Roch coefficient of the big class are both approximated by those of Kähler classes on higher compactifications.

Let S^{n+1}_{+} be the open hemisphere of the unit sphere S^{n+1} centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k {alpha} and the critical regime beta=1+k{alpha}, where beta is the growth order of the profile at the origin, f(r) almost equals r^{beta} as r descends to 0. Under the structural condition that f^{1/(1+k{alpha})} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_{infty}>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.

This paper investigates the canonical Koszul one-form at the interface of Jean Marie Souriau's Lie group thermodynamics and Paulette Libermann's symplectically complete foliations to describe geometric dissipation. On the Poincaré half-plane $\mathbb{H}$ under $SL(2, \mathbb{R})$, the equivariant momentum map level sets form a regular Libermann foliation framing the conservative boundaries. The Koszul-derived dissipative vector field acts strictly transverse to these leaves, driving a metriplectic flow that breaks Noether conservation laws and generates entropy. Quantum wise, we develop Berezin quantization on para-Kähler symmetric spaces, where the Koszul potential acts as a weight determining state density via the Fisher Souriau metric, and formalize Kostant's geometric quantization using the real dual polarizations of a bi-Lagrangian web.

We prove that closed Einstein four-manifolds $(M,g)$ with positive sectional curvature and Euler characteristic and signature satisfying $2\chi(M)-3|\tau(M)|\le4$ must be homothetically isometric to the round $S^4$ or to $\mathbb{CP}^2$ with the Fubini-Study metric. The proof relies on an upper bound on the ADM mass of conformal blow-ups of $(M,g)$, combined with a lower one from \cite{GM}.

We study third-order linear differential equations through their associated Pfaffian and projective foliations. In the regular-singular setting, we introduce a Fuchs-adapted Pfaffian model in which the indicial polynomial appears directly in the special-fiber singular scheme, and Frobenius resonance acquires a projective geometric interpretation.

We prove that the energy functional associated with surfaces of prescribed mean curvature in $\mathbb{R}^3$ exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form $H \cdot \text{vol}_{\mathbb{R}^3}$ has a primitive satisfying certain bounds. Under milder assumptions on $H:\mathbb{R}^3 \to \mathbb{R}$, we show that an analogous convexity estimate holds along the heat flow of the functional (the $H$-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the $H$-surface flow with small-energy initial map of class $C^0 \cap W^{1, 2}$ on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the $H$-surface system) with the same boundary value.

Let $X: M^{n}\to \mathbb{R}^{n+1}$ be a complete self-shrinker with constant squared norm of the second fundamental form $S$. In this paper, we prove that if $S\leq \frac{10}{7}$, then $S=1$ or $S=0$, and the self-shrinker is isometric to either a round sphere $\mathbb{S}^n(\sqrt{n})$ with the center at the origin, or a cylinder $\mathbb{S}^k(\sqrt{k})\times \mathbb{R}^{n-k},~~1\leq k\leq n-1$, or a plane through the origin.

We prove that the product of three connected noncompact smooth manifolds without boundary admits a complete metric of uniformly positive scalar curvature. The proof uses the additivity of Morse indices on products and an orientation-free version of Das's open Morse–surgery construction. We also give an example of two open manifolds whose product admits no complete metric of nonnegative scalar curvature.

We prove that plane waves are the only gravitational pp-wave spacetimes which are distinguishing; in particular, they are the only strongly causal ones. This is used to prove a null version of Anderson's rigidity theorem, conjectured by Costa e Silva and Flores: every geodesically complete, transversally complete, strongly causal, Ricci-flat Brinkmann spacetime of dimension $4$ is isometric to a quotient of a plane wave spacetime.

Given a Riemannian manifold $(M,g)$ equipped with a positive density function $f = e^{-\varphi}$, we study the weighted sectional curvature $\overline{\sec}_\varphi(U,V)$ and the $k$-th intermediate weighted Ricci curvature $\overline{\text{Ric}}_{k,\varphi}(U,V)$. We show that $\overline{\sec}_\varphi(U,V)$ and $\overline{\text{Ric}}_{k,\varphi}(U,V)$, together with the weighted second fundamental form $\widetilde{\mathrm{I\!I}}$, play central roles in controlling the topology of the manifold. Using the theory of $\widetilde{C}(k)$ functions, variational methods, and Morse theory, we prove that by imposing suitable restrictions on these quantities, an $n$-dimensional compact manifold with boundary or a proper open $n$-dimensional manifold is diffeomorphic to an $n$-ball or to a certain CW complex. This shows that such rigidity phenomena is not confined in the unweighted case, but persist in the weighted setting as well. Lastly, we prove a weighted analogue of Frankel's Theorem under positive intermediate weighted Ricci curvature. These results extend classical theorems from the unweighted setting to the weighted case, demonstrating that $\overline{\sec}_\varphi(U,V)$ and $\overline{\text{Ric}}_{k,\varphi}(U,V)$ provide effective tools in Riemannian geometry.

This paper concerns the geometry of gravitational pp-wave spacetimes. We give a counterexample to the Ehlers–Kundt conjecture. In fact, we show that geodesic completeness is generic, in the sense of Baire category, among gravitational pp-waves. We also define a concept of “universal pp-wave”, i.e. a pp-wave whose metric approximates that of every other gravitational pp-wave arbitrarily closely on arbitrarily large compact sets, and we show that there are geodesically complete universal pp-waves. Our machinery also suffices to give a new proof of the polynomial case of the Ehlers–Kundt conjecture (previously proved by Flores and Sánchez).

Let $(M, g)$ be an $n$ $(\ge 3)$ dimensional, non-collapsed compact Riemannian manifold and $\operatorname{Ric}^-$ be the negative part of the Ricci curvature and $\beta \in (\frac{2n}{n+2}, 2)$. We prove a relative volume comparison theorem when $|\operatorname{Ric}^-|^\beta$ is in the Kato class (cf. Definition 1.1), which results from a new integral Laplace comparison theorem in the spirit of \cite{PW} for a suitable conformal metric. This partly addresses an expectation in \cite{TZZZZ}, where the same result was proven when $|\operatorname{Ric}^-|^2$ is in the class.