Papers

We prove two constant rank theorems for the Ricci tensor in two geometric settings: manifolds with harmonic curvature and gradient Ricci solitons. In both cases, if the $k$-intermediate Ricci curvature satisfies $\Ric_k\geq0$ for some $1\leq k\leq n-2$, then the Ricci tensor has constant rank and its nullity distribution is parallel. In the complete case, the universal cover splits isometrically along the Ricci-null directions. We also show that the range $1\leq k\leq n-2$ is sharp: in every dimension $n\geq4$, nonnegative Ricci curvature alone does not imply the constant rank property in either setting.

Let $(M,g)$ be a compact Riemann surface with smooth boundary $\partial M$. Let $K(x)>0$, $h(x)> 0$ or $h(x)\equiv 0$ be two smooth functions on $M$ and $\partial M$ respectively. In this paper, we consider the following double mean field type equation with a nonlinear Neumann boundary condition related to a prescribed curvature problem \begin{equation}\label{equation:01} \begin{cases} \displaystyle -\Delta_g u+\frac{4\pi}{|M|} =2\rho\frac{Ke^u}{\int_MKe^u\,dv_g}, & \text{in }M,\\[1.2em] \displaystyle \partial_\nu u =2\rho'\frac{he^{u/2}}{\int_{\partial M}he^{u/2}\,ds_g}, & \text{on }\partial M, \end{cases} \end{equation} where $\rho> 0,\ \rho'\geq 0$ and $\rho+\rho'=2\pi$, $\nu$ is the outward unit normal vector to $\partial M$. When $h\equiv 0$, the second equation changes to $\partial_\nu u=0$ on $\partial M$. By introducing a boundary integration constant, we obtain a sufficient condition under which \eqref{equation:01} admits a solution. As a geometric application, we solve the prescribed curvature equation for both the Gauss curvature and the geodesic curvature in the case that $M$ is topologically a disk. When $(M,g)$ is the standard hemisphere $\mathbb S^2_+$, let $p\in \partial \mathbb S^2_+$ be a maximum point of $$ \Phi(q)=h(q)+\sqrt{h(q)^2+K(q)}, \qquad q\in\partial \mathbb S^2_+. $$ Suppose \begin{equation*} K_\nu(p)+2\Phi(p)(-\Delta_{\partial \mathbb S^2_+})^{1/2}h(p)<0. \end{equation*} Then the prescribed curvature problem \begin{align*} -\Delta_g u+\frac{4\pi}{|M|}=2Ke^u, \text{in }\mathbb S^2_+,\ \ \partial_\nu u=2he^{u/2}, \text{on }\partial \mathbb S^2_+. \end{align*} admits a smooth solution.

We consider the cone Yang-Mills functional recently introduced by Tseng and Zhou. This functional generalizes the Yang-Mills functional by coupling it to a closed 2-form which can represent a geometric structure of interest, e.g. a symplectic form or Ricci form. While the perspective of dimensional reduction is not always possible, we show that a cone Yang-Mills field is always locally the dimensional reduction of a pure Yang-Mills field. We then prove a monotonicity formula and use the dimensional reduction framework to prove an $\epsilon$-regularity theorem in two ways. Finally, we prove a compactness theorem for sequences of cone Yang-Mills connections with bounded action and $L^p$-bounded Higgs fields. Away from a singular set of codimension at least 4 and after suitable bundle identifications, the sequence subconverges in the $C^\infty_{\operatorname{loc}}$-topology.

Let $(M^n,g_0)$ be a complete Riemannian manifold with $\Ric_{g_0}\ge-(n-1)g_0$ whose local universal covers are uniformly $\delta$-Reifenberg-close to $\R^n$ below a fixed radius $\rho$. We show that for every $\alpha\in(0,1)$ there are $\delta(n,\alpha)>0$ and $T(n,\rho,\alpha)>0$ such that, if $\delta\le\delta(n,\alpha)$, a complete Ricci flow $g(t)$ with $g(0)=g_0$ and $|\Rm_{g(t)}|\le\alpha/t$ exists on $(0,T]$. On closed manifolds this is a uniform lower bound for the lifetime of the Ricci flow; on noncompact manifolds, where the initial curvature may be unbounded, it is an existence theorem, and the flow is unique among complete Ricci flows from $g_0$ with a curvature bound of the form $A/t$. The key step is a finite-buffer comparison of two Ricci flows through the Ricci–harmonic map heat flow, following Lee, which transfers a Type-I curvature bound from a complete selected flow to the incomplete flow lifted to a local cover. Consequently, below a dimensional Reifenberg threshold, collapsing closed manifolds admit singular nilpotent fibrations over their Gromov–Hausdorff limits. The Reifenberg hypothesis cannot be weakened to noncollapsing of the local covers: products of shrinking circles with Nifa's four-manifolds have $\Ric\ge0$ and uniformly noncollapsed local covers, but admit no continuous Gromov–Hausdorff approximations onto their limit.

We prove that, on any compact Riemann surface, there are only finitely many meromorphic projective connections with prescribed singular points and real exponent differences whose projective monodromy is unitarizable. We also establish properness and growth estimates for holonomy, including a relative version for smooth marked families, and derive finiteness results for spherical conical metrics and ramification divisors of stable extensions.

We compute the index and nullity of any harmonic morphism from $S^3$ to $S^2$.

Ideal Poisson–Voronoi tessellations are random tessellations that arise as limits of Voronoi diagrams for Poisson point processes as their intensity tends to zero. For the Riemannian metric on a symmetric space of higher rank, Frączyk, Mellick, and Wilkens (arXiv:2307.01194) proved that the corresponding ideal Poisson–Voronoi tessellation is almost surely completely connected, with an unbounded side between any two cells. This contrasts sharply with the geometry of ideal Poisson–Voronoi tessellations in hyperbolic spaces studied by D'Achille, Curien, Enriquez, Lyons, and Ünel (arXiv:2303.16831). In this paper we introduce and study ideal Poisson–Voronoi tessellations for Finsler metrics on higher rank symmetric spaces. We show that ideal Poisson–Voronoi tessellations with respect to polyhedral Finsler metrics can exhibit either behavior seen in these previous works: generically they resemble the hyperbolic case, where almost surely all cells have bounded sides. We further introduce the ideal Selberg–Poisson–Voronoi tessellation of the space of positive definite symmetric matrices, which has particularly nice properties.

We study whether an unnormalized graph Laplace operator can determine both the geometry of a manifold and its sampling density. On a fixed compact smooth manifold without boundary, we use a known positive, continuous, strictly decreasing radial kernel at a fixed positive bandwidth. For positive sampling densities, the operator built from geodesic distances determines the Riemannian metric and identifies the density up to equality almost everywhere. When the metric is induced by a Euclidean embedding, the operator built from ambient distances also determines the embedding up to rotations, reflections and translations, and the density. On compact parameter classes with positive continuous densities, minimizing the distance between empirical and popula- tion operators gives estimates that converge to the true parameters under independent, identically distributed sampling. This recovery result assumes that the empirical operator can be evaluated at arbitrary points of the manifold.

We develop an intrinsic rigidity theory for bar-and-joint frameworks whose vertices are constrained to a Riemannian manifold and whose edges are given by minimizing geodesic arcs. Such frameworks are called rigid if they cannot be deformed beyond the ambient isometries, while preserving the geodesic edge lengths. For admissible placements, for which adjacent vertices are joined by unique minimizing geodesics, we define a coordinate-free rigidity operator as the differential of the squared geodesic distance of the edges. By the first-variation formula, its edge rows are expressed in terms of the Riemannian logarithm, providing an intrinsic analogue of the classical Euclidean rigidity matrix. Trivial motions are characterized by orbits of the diagonal action of the isometry group, leading to notions of infinitesimal and local rigidity. We further develop a covariant second-order rigidity theory using the Riemannian Hessians of the edge constraints and characterize second-order rigidity through self-stresses and associated quadratic forms. Finally, we show that infinitesimal and second-order rigidity can be used as sufficient conditions for the local rigidity of an intrinsic framework. Our overarching goal is to provide a unified differential-geometric formulation of intrinsic framework rigidity on smooth manifolds.

Real data are often arbitrary, indirect observations of an unknown underlying process, and those observations can distort its geometry. Existing geometric methods equip data with the induced geometry of the ambient space, but this entangles the true geometry of the process with the distortion of the observation. To address this, we introduce Intrinsic Geometry Matching (IGM): a scalable approach to learn the intrinsic geometry of the underlying process. Specifically, we learn the carré du champ operator of the process, allowing us to recover its diffusion geometry. Our method uses the fact that data often come with natural perturbations, like steps in a time series, to recover the intrinsic carré du champ while transforming naturally with changes in observation: it is observation-equivariant. Furthermore, since this equivariance relates the geometry of different spaces, we can train a model using additional sources of data, even when that data lies in another space. We evaluate IGM on molecular dynamics, image, and video data and find better performance and scaling than existing neural and graph-based geometric tools.

We study compact static manifolds with boundary from a spectral point of view, with particular emphasis on the relation between Dirichlet, Neumann, and Robin spectra and the nodal structure of static potentials. We establish comparison inequalities linking Dirichlet and Neumann eigenvalues with the scalar curvature and derive sufficient spectral conditions ensuring that a static potential changes sign. We also obtain bounds on the number of connected components of the zero set, sufficient conditions for its connectedness, and sufficient conditions ensuring that its components intersect the boundary. Finally, we study variational properties of Robin static triples and characterize compact Robin static triples whose metrics are critical for the first mixed Robin–Dirichlet eigenvalue under admissible volume-preserving deformations.

We study the determinant line bundle for a family of elliptic operators on compact manifolds with boundary with the associated Quillen metric and Bismut-Freed connection, and we prove the global anomaly formula in this context. This extends Witten's holonomy theorem to manifolds with boundary. Witten studied the limiting behavior of the eta invariant of a Dirac operator on the total space of a fibration over $\mathbb{S}^1$ when the metric in the base $\mathbb{S}^1$ is blown up in a certain manner, a process known as taking the adiabatic limit. In this paper, we construct the calculus of adiabatic $b$-pseudodifferential operators, which naturally synthesizes the $b$-calculus and the adiabatic calculus of pseudodifferential operators. Using this calculus, and assuming the fiber operators to be invertible, we compute the adiabatic limit of the natural eta invariant in this setting and relate it to the integral of the Bismut-Freed connection and new correction terms coming from the boundary of the fibers.

We prove that every Lawson surface $\xi_{k,l}\subset\mathbb S^3$ of genus $kl>1$ is non-algebraic: any real polynomial vanishing on $\xi_{k,l}$ vanishes identically on $\mathbb S^3$. This answers a question posed by Hsiang in 1967 and proves Lawson's 1970 conjecture for $\xi_{2,2}$. The proof uses a reflection principle for the monodromy of the associated family of flat connections.

The purpose of this note is twofold. First, we revisit the classical regularity theory of harmonic maps in the setting of maps defined on a lattice $\mathbb{Z}^d$ instead of $\mathbb{R}^d$ or some other manifold. The target space of our harmonic maps will be spheres $\mathbb{S}^{N-1}, N\geq 2$. A harmonic map $\sigma : \mathbb{Z}^d \to \mathbb{S}^{N-1}$ is a map which in any finite subgraph $\Lambda$ minimizes the internal Dirichlet energy $\frac 1 2 \sum_{i \sim j \in \Lambda} \|\sigma_i -\sigma_j\|^2$ given the boundary condition $\sigma_{|\partial \Lambda}$. One key difference between our setting and the continuum is that our main Hölder regularity statement in a large domain $\Lambda\subset \mathbb{Z}^{d=2}$ does not require any regularity assumption on the map $\sigma$ restricted to the boundary of the domain. For example, when $d=2$ and $N=3$, the Hölder regularity induced in the bulk of the domain holds uniformly in all boundary configurations $\sigma \in (\mathbb{S}^{2})^{\partial \Lambda}$ (this is in contrast with the continuum which requires extensions in suitable Sobolev spaces). The fact we obtain results uniformly in the boundary conditions allows us to extend our results for positive temperatures. Namely, as $T \searrow 0$, we obtain a local regularity theory for all infinite volume Spin $O(N)$ Gibbs measures, at least when $d=2$ and $N\geq 3$. We also discuss implications of our analysis in dimensions $d\geq 3$.

Let $M$ be a connected Hamiltonian stationary Lagrangian surface of constant Gaussian curvature $K$ in a two-dimensional complex space form $\Mtilde^2(4\epsilon)$ of constant holomorphic sectional curvature $4\epsilon$, and let $H$ denote its mean curvature vector field. We prove that either $K=\epsilon$ or $K=0\ne\epsilon$ and $|H|$ is constant. In particular, if $|H|$ is non-constant, then $K=\epsilon$.

For polarized manifolds, we show that semistability for the quantized Mabuchi energy at one fixed finite parameter is equivalent to uniform K-stability and to unique existence of constant scalar curvature Kähler (cscK) metrics. We obtain sharp comparisons between quantized Mabuchi slopes at different parameters and with the ordinary Mabuchi slope. In the presence of automorphisms, we give an equivariant criterion using torus twists under the vanishing of the Futaki character. The estimates also hold for finite-energy geodesic rays in compact Kähler classes. On a fixed ruled surface with discrete automorphism group, we construct cscK polarizations with unbounded critical parameters. Their ratios to the $\alpha$-invariant are also unbounded.

We prove that the singular locus of the geodesic flow on the unit tangent bundle of a nonpositively curved surface enjoys an additional regularity property: its characteristic function belongs to the geometric anisotropic Sobolev space $H^{5/8-}_xH^{-13/16}_\theta$. In particular, this implies that if the geodesic flow is not ergodic with respect to the Liouville measure, then the zero set of the curvature must be thick in the sense that it supports a nonzero, nonnegative function in $H^{5/8-}$. The proof relies on semiclassical analysis and a new radial estimate for parabolic flows.

We determine the minimal observation time for the heat equation on a closed almost-Riemannian surface, for a class of observation sets leaving sufficiently thin collars around simple Baouendi-Grushin circles. The structure and density are analytic near these circles, and the coefficients may depend on both variables. The formula is expressed through a complex phase describing concentration near the singular set and extends the thresholds known for separated models. Regular unobserved regions away from the circles do not change this time, provided all other singularities are observed. For singular circles of constant bracket step greater than two, local analyticity implies failure of observability at every time for observation sets at positive distance. We also construct quasimodes concentrated near simple singular circles, with exponentially small errors. Their superpositions give a lower bound for the Schrödinger observation time.

We study Ricci pinching problem on complete Kähler manifolds. For complete noncompact manifolds with Euclidean volume growth, we prove flatness under nonnegative bisectional curvature and Ricci pinching, and Ricci flatness under nonnegative Ricci curvature, quadratic curvature decay, and the same pinching condition. In both cases, the pinching constant may tend to zero at a logarithmic rate. Examples show that the logarithmic exponent is sharp in complex dimension two. Second part of the paper proves that a complete Kähler manifold of complex dimension at least two with positive holomorphic bisectional curvature is compact if its sectional curvature is nonnegative and its Ricci curvature is uniformly pinched outside a compact set. No volume-growth assumption or curvature upper bound is needed. Heat smoothing of the Busemann function supplies a strictly plurisubharmonic exhaustion with bounded gradient, so the general top Ricci-mass theorem of Datar–Pingali–Seshadri applies. The new ingridents are integral estimates for the Ricci form. The analytic arguments use scalar curvature gap theorems and a critical-$L^n$ vanishing theorem for closed real $(1,1)$-forms under nonnegative Ricci curvature and nonnegative quadratic orthogonal bisectional curvature. The Hodge-Laplace heat eqution deformation conserves the signed top-degree integral, including its initial value, while norm domination and scalar heat-semigroup decay force this integral to vanish. For semipositive initial forms this gives pointwise vanishing of the top wedge power.

In \cite{ThurstonStretch} Thurston introduced stretch paths and stretch maps between hyperbolic surfaces. In this paper we extend similar constructions to the space of convex real projective structures on surfaces. Using Fock-Goncharov-like parameters, we introduce stretch paths and construct explicit stretch maps between the projective surface. We show that stretch maps are $e^t$–Lipschitz with respect to a triangular metric we introduce on the projective surface. This allows us to gain control on several length functions on an open set of the space of convex real projective structures (those where there is no “local back–tracking"). As an application we obtain pointwise upper bounds for the Hilbert entropy of some convex real projective structures in terms of Fock-Goncharov-like parameters.

We study the asymptotic behaviour of the nonlocal perimeter of tubular neighbourhoods. For a closed $C^{1,1}$ submanifold of Euclidean space, we show that a suitable rescaling of the nonlocal perimeter of its tube converges to an explicit constant times the Hausdorff measure of the submanifold. The constant is the fractional perimeter of the unit ball in the codimension of the submanifold. For finite point sets, we obtain an exact tube formula in terms of a generalised hypergeometric function. In contrast with the classical Weyl formula, this expansion has infinitely many terms, contains both noninteger and integer powers of the tubular radius, and has coefficients that depend on the relative positions of the points rather than only on their intrinsic geometry. We also establish a point-set expansion for general analytic kernels.

An isometric immersion $X: \Sigma^n \longrightarrow \mathbb{E}^{n+1}$ is biharmonic (BHH) if $\Delta^2 X = 0$, i.e., if $\Delta H =0$, where $\Delta$ and $H$ are the metric Laplacian and the mean curvature vector field of $\Sigma^n$ respectively. More generally, biconservative hypersurfaces (BCHs) are isometric immersions for which only the tangential part of the biharmonic equation vanishes. BCHs are first described and then constructed.

This paper considers a four-dimensional Riemannian manifold with an additional circulant tensor structure $Q$. We prove that such a manifold is almost-Einstein if the curvature tensor preserves the action of the structure on the last two arguments. Additionally, we examine an associated Riemannian product manifold $(M, g, Q)$ where $P=Q^{2}$, deriving explicit expressions for the curvature tensor in terms of the metric and the product structure. The sectional and Ricci curvatures of these manifolds are determined using special bases generated by the images of a given vector under the structure $Q$. Finally, we construct a non-trivial example of a four-dimensional locally product manifold by embedding a four-dimensional hypersurface into a ten-dimensional Euclidean space.

We study the exponential map on the group of real two-by-two matrices with determinant one, endowed with a left-invariant Riemannian metric. Using the Euler-Poincaré equation and its linearization, we determine the conjugate and Maxwell times and identify a domain on which the exponential map is a global diffeomorphism onto the complement of the Maxwell stratum. This provides an alternative proof that the first Maxwell time coincides with the cut time. We obtain an explicit matrix description of the cut locus and characterize the globally minimizing geodesics. Furthermore, we analyze the first Maxwell-time function, determine its unique global minimum, and show that the injectivity radius is $\frac{\pi \sqrt{14}}{4}$. Finally, we characterize the intersection of the cut and conjugate loci and prove that every geodesic is equi-optimal.

We establish a family of sharp higher order isoperimetric type inequalities for smooth compact surfaces in Euclidean spaces of arbitrary codimension. These inequalities involve arbitrarily high derivatives of the boundary curve, together with the mean curvature vector of the surface. For example, the first and second order inequalities yield the sharp two-sided estimate for the isoperimetric deficit: \begin{align*} & \frac{\pi}{R}\int_{\partial M}|x-R\nu|^2\, ds +2\pi\int_M x\cdot H\, dA \le L^2-4\pi A\\ \le& \frac{\pi}{R}\int_{\partial M}|x-R\nu|^2\, ds +2\pi\int_M x\cdot H\, dA +\frac{\pi}{3R}\int_{\partial M}|x+R^2\kappa|^2\, ds, \end{align*} where $R=\frac{L}{2\pi}$ and the origin is chosen to be the centroid of the boundary. For minimal surfaces, equality in the lower bound holds precisely for flat disks, whereas the second inequality admits non-round equality cases. Our results extend that of Carleman, Sachs, Chakerian, and Li-Schoen-Yau. A distinctive feature of our work is the unexpected rigidity structure arising at higher orders. At the first and second orders, equality curves admit decompositions into circular Fourier modes supported in mutually orthogonal planes. Remarkably, this phenomenon fails at higher orders. We establish a multiplicative factorization of the unit tangent map into elementary planar rotations acting on a fixed unit vector, providing a characterization of equality cases at arbitrary order. As a result, we construct a third order equality curve in $\mathbb R^3$ whose Fourier modes cannot be decomposed into mutually orthogonal circular modes.

On a complete noncompact Kähler manifold $M^n$ of complex dimension $n$ with nonnegative Ricci curvature, we prove that polynomial growth holomorphic functions of degree $d$ ($d\geq 0$) have an dimension upper bound $C(n)(1+d)^n$. In particular, $\dim_{\C}\OO_d(M)\leq C'(n)d^n,$ for $d\geq1$, and the estimate is sharp.

We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For $L^2$-normalized local solutions at frequency $\lambda$, cubic integrals against smooth compactly supported weights tend to zero whenever the curvature is nonzero almost everywhere where the weight is nonzero. Finite-order nondegeneracy yields sharp polynomial decay rates. In particular, on closed surfaces with nowhere-vanishing curvature, we obtain the rate $O(\lambda^{-1/6})$, attained by spherical harmonics on the round sphere. This proves the cubic case of the Gaussian moment conjecture for all normalized real eigenfunctions on closed negatively curved surfaces. The local proof rests on an exact identity relating the reduced phase at a Fermat point to Gaussian curvature. On complete hyperbolic surfaces with positive systole, we also prove global $O((\log\lambda)^{-3/2})$ decay for $L^2$-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.

We generalize Moser's stability theorem for minimal hypersurface foliations to arbitrary codimension. Let $\mathbb{T}^N=\mathbb{R}^N/\mathbb{Z}^N$ be the standard torus, and let $P\subset\mathbb{R}^N$ be an $m$-plane satisfying a Diophantine condition relative to $\mathbb{Z}^N$. We prove that, under every sufficiently small smooth $\mathbb{Z}^N$-periodic perturbation of the Euclidean metric, the affine foliation of $\mathbb{T}^N$ induced by $P$ persists as a smoothly conjugate foliation by minimal $m$-dimensional submanifolds. The normalized conjugating displacement depends smoothly and tamely on the metric and is locally unique. The proof uses Hamilton's formulation of the Nash-Moser implicit function theorem.

We prove that round circles $C_L$ are the unique minimisers of integral Menger curvature among embedded closed $C^1$ curves $\gamma$ of prescribed length $L$ for every $p\geq 2$, i.e. $M_p(C_L) \leq M_p(\gamma)$, with equality if and only if $\gamma$ parametrises a round circle. The proof combines a three-point symmetrisation, an analysis of diagonal truncations, and the periodic Wirtinger inequality. For $p>3$, known self-avoidance and regularity results extend the minimisation result to closed rectifiable curves without assuming embeddedness or smoothness, provided the length of the curve image equals $L$. In particular, $M_p$ detects the unknot and is basic for $p>3$, thereby answering a question posed repeatedly in the literature.

We prove several regularity theorems concerning Lipschitz viscosity solutions to the minimal surface system. The main lemma (i.e. “Decoupling Lemma") says that the component functions of such solutions satisfy the homogeneous Pucci inequalities. This allows us to apply the regularity theory for scalar viscosity solutions for elliptic PDE component-wise to a viscosity solution for the system. As a result of the Decoupling Lemma, we deduce that 2D Lipschitz viscosity solutions to the system are analytic in the interior, and obtain a quantitative partial regularity theorem. We also prove a geometric Harnack inequality for Lipschitz solutions. The Harnack inequality yields a strong maximum principle and half-space Liouville theorem. Another significant consequence of the lemma is a boundary regularity theorem for Lipschitz viscosity solutions. Notably, our theorem does not require any density assumption at the graph boundary, and we do not assume stationarity. This is used to prove a classification theorem for the singular set of a 4D stationary solution on a bounded domain. We also prove an $\varepsilon$-regularity theorem extending a result of the author. To conclude, we survey open problems. As an example, we have included a Bernstein-type theorem for 3D homogeneous entire Lipschitz viscosity solutions. Taken together, our results suggest that viscosity methods are a useful tool for studying the regularity of minimal graphs.