Papers

Let $\Lambda$ be the Dirichlet-to-Neumann operator on the boundary of a compact three-dimensional Riemannian manifold. Using the Lee–Uhlmann full-symbol formula and Weyl's invariant theory, we compute the Guillemin–Wodzicki residues of $\Lambda$ and $\Lambda^2$ explicitly, and hence the first two logarithmic coefficients in the Steklov heat trace. As an application, we prove that geodesic balls in simply connected three-dimensional space forms are determined by their Steklov spectra among smooth domains in the same space form. More generally, we obtain a rigidity theorem in the class of compact constant-curvature manifolds with smooth, not necessarily connected, boundary. In the Euclidean case, we also prove spectral uniqueness for concentric spherical shell regions.

In this paper, we establish the Weinstock inequality for the first non-zero Steklov eigenvalue on star-shaped mean convex domains in hyperbolic space $\mathbb{H}^n$ for $n\geq 3$. We note that when $n\geq 4$, the result was obtained in our previous paper [23]. In particular, when the domain is convex, our result gives an affirmative answer to Open Question 4.27 in [13] for the hyperbolic case.

We prove sharp Weinstock inequalities for the first nonzero Steklov eigenvalue of smooth outward-minimizing domains in Euclidean space and hyperbolic space. The method is based on the weak inverse mean curvature flow of Huisken–Ilmanen. The main new ingredient is an endpoint distributional monotonicity argument obtained from the calibrated weak formulation and the Gauss–Green formula for divergence-measure fields.

We characterize minimizers in Gamow's liquid drop problem.

It has been recently shown that the notoriously divergent $L ^2 $ metric on the moduli space of hyperbolic monopoles can be made finite by the introduction of a modified gauge fixing condition \cite{franchetti:2024}. In this paper we compute this modified $L ^2 $ metric for the mass $\tfrac{1}{2}$ 2-monopoles. The resulting metric is actually a complex non-degenerate bilinear form, which restricts to a Riemannian metric on the 4-dimensional subspace of inversion symmetric 2-monopoles. Remarkably, we have been able to compute the $L ^2 $ form explicitly in terms of elementary functions and elliptic integrals. Its asymptotic form matches what was expected on the basis of previous results in the literature.

We study the geometry of simplicial cochains on Vietoris-Rips complexes built from i.i.d. samples of a compact embedded manifold. By integrating differential forms over affine simplices, we define a cochain map from smooth forms to simplicial cochains and equip the latter with kernel-weighted inner products determined by ambient pairwise distances. At scales where the sampled complex recovers the homotopy type of the manifold, we prove quantitative high-probability convergence of these inner products and, under uniform sampling, of the associated codifferential energies to their continuum counterparts. As consequences, we obtain spectral upper bounds for the discrete Hodge Laplacian and, for orientable manifolds, convergence of harmonic representatives of discretized harmonic one-forms and consistency of harmonic smoothing for circular coordinates.

Let $K\subset\mathbb{R}^n$ be a convex body and $C=\frac12(K+(-K))$ its central symmetral. We study the defects $\mathcal A_k(K)=W_{n-k}(C)-W_{n-k}(K)$ through the even–odd decomposition of the support function. We derive exact even mixed-volume expansions, identify their Krawtchouk transform with the mixed difference-body coefficients, and obtain a Kubota formula expressing each defect as the averaged symmetrization gain of projections. These identities yield strengthened isoperimetric inequalities. The quadratic defect is the average of the absolute oriented areas of projected Wigner caustics and obeys a sharp spherical-harmonic stability estimate. We give explicit formulas in dimensions $2$, $3$, and $4$. The four-dimensional expansion includes a sign-indefinite quartic Wigner volume; on a plane of cubic harmonics we find nonspherical constant-width bodies for which it vanishes. Finally, we prove a projection-transfer bound and an arbitrary-dimensional expansion for constant-width bodies.

In this note, we prove that there exist two families of bi-rotational asymptotically conical self shrinkers with boundary called `trumpets', and two families of bi-rotational self shrinkers with boundary which close up called `disks', so that these families together with two generalized self shrinking cylinders, and a minimal cone foliate the whole space except some compact set containing the origin.

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n +m \geq 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\textbf{singular minimal cone as the asymptotic model}$.

In this paper, we study compactness, rigidity, and related geometric properties of complete Kähler Ricci shrinkers through the polarized Fano fibration structure.

We initiate a study of centrally extended ($\textit{anomalous}$) symmetries in Kähler geometry, focusing on the simplest, and most ubiquitous, Abelian case. In particular, we provide a local description of the geometry admitting such isometries. A long-standing no-go theorem asserts that there is an obstruction to gauging such symmetries in the purely Kähler framework. Utilizing the language of sypersymmetry, we then show that these symmetries may be gauged within the setup of generalized Kähler geometry. Our results may be applied to quotients, T-dualities, etc.

We prove the Weinstock inequality for the first Steklov eigenvalue of convex domains in hyperbolic space $\mathbb{H}^{n}$, resolving Open Question 4.27 of Colbois-Girouard-Gordon-Sher(2024) for the remaining case $n=3$. Our argument replaces the global monotonicity required in earlier work Gu-Li-Wan(2025) with a one-crossing property, which is established via an explicit slope comparison. The proof works uniformly for all $n\geq 3$.

Anatomical image registration commonly relies on a sequential pipeline where an affine alignment is estimated first and then held fixed while a non-rigid diffeomorphic deformation is applied. This two-step process often leads to suboptimal results, as the initial stage can absorb local deformations, biasing the residual passed to the diffeomorphic registration. To address this, we introduce a Joint Affine-Diffeomorphic framework, based on the large deformations model, that estimates both global affine and local diffeomorphic motions simultaneously within a single optimization. We propose two models: a Full Affine (FA) model that combines affine and diffeomorphic deformations, and a Decomposed Affine (DA) model that restricts the affine part from FA to rotations, translations, and anisotropic scalings. To numerically implement these models for image registration tasks, we develop a tailored optimization strategy that combines progressive affine enrichment, gradually increasing the complexity of the affine component, with a variational weighting scheme that smoothly manages the coarse-to-fine handover between the affine and diffeomorphic components. Evaluated on 2D synthetic datasets and 3D brain MRIs from the IXI cohort, our unsupervised approach avoids the pathological deformations of sequential baselines. We demonstrate that our joint formulation outperforms in Dice overlap two state-of-the-art deep learning foundation models, CARL and uniGradI-CON, as well as a sequential baseline using FLIRT for the classical affine registration followed by LDDMM. Our implementation is publicly available.

We introduce fully nonlinear curvature flows of horo-convex hypersurfaces in the sphere that preserve an arbitrarily prescribed spherical quermassintegral. The non-local normalization is defined relative to a fixed ambient origin, and we prove that the evolution is governed by a single smooth fixed-origin equation for all time. For monotone, homogeneous curvature functions satisfying concavity and inverse concavity, we establish preservation of horo-convexity, uniform curvature pinching, a direct estimate for the non-local coefficient. A Tso-type argument then yields global curvature bounds and long-time existence. We further prove exponential decay of the traceless second fundamental form and exponential $C^\infty$ convergence to the geodesic sphere centered at the fixed origin whose radius is determined by the preserved quermassintegral.

We prove that, for every $n\ge 2$, every positive entire solution of the critical CR Yamabe equation $4\Delta_b u = n^2 u^{(Q+2)/(Q-2)}$, $Q=2n+2$, on the Heisenberg group $\mathbb H^n$ is a Jerison-Lee bubble. No integrability, decay, boundedness, or symmetry is assumed. Together with the theorem of Catino, Li, Monticelli, and Roncoroni in $\mathbb H^1$, this classifies the positive entire solutions in every dimension. Both Euclidean routes to such a statement lose their starting configuration here. Hyperplane reflections are not CR automorphisms, and a CR inversion preserves its Koranyi sphere only setwise, so the difference between a solution and its Kelvin transform need not vanish on the sphere one inverts in. Nor is there a substitute a priori bound to fall back on: the only scale-invariant estimate available for every positive solution is a critical Morrey bound, which concentration saturates and which yields neither decay nor a small-mass regularity principle. We proceed instead from two exact consequences of the Green representation, which every positive solution is shown to satisfy. Reciprocity with a bubble $U$ converts the distance from $U$ into a nonnegative convex deficit, so that no information about the sign of a linearized quadratic form is required; differentiating the same reciprocity along the conformal orbit of $U$ gives a nonlinear barycentre identity, which forbids that deficit from concentrating at a single point of the CR sphere. Together these isolate the bubble manifold in a class carrying no energy bound. Quantizing the Jerison-Lee tensor defect against this isolation, and alternating the resulting budget with the Morrey bound, drives the energy-growth exponent into the range where the defect must vanish. Both identities use only conformal covariance and an exact positive Green representation.

The space of $W^{2,2}$-isometric immersions of a surface into $\mathbb{R}^3$ arises naturally in the variational theory of thin elastic sheets: it is precisely the finite-bending class, where the bending energy — the $L^2$-norm of the second fundamental form — is finite. For sheets with negative Gaussian curvature, previous work has identified branch points, where "too many" asymptotic directions meet — or, equivalently, where the index of the Gauss map is not $-1$ — as a potentially important mechanism in shape selection and pattern formation. Such branch points are precluded for $C^2$-isometric immersion. We show that this index-based notion of branch points extends to the full finite-bending class: Namely, for every $W^{2,2}$ isometric immersion of a negatively-curved surface, the index of the Gauss map is well-defined at every point, and the set of branch points is discrete. We further show that the index is stable under $W^{2,2}$-convergence, and thus, an isometric immersion with branch points cannot be approximated by $C^2$-isometric immersions. Conversely, we show that every negatively-curved metric locally admits $W^{2,2}$-isometric immersions (in fact, $C^{1,1}$) with branch points of arbitrary order. Consequently, $C^2$-isometric immersions are, in general, not dense among finite-bending ones, in stark contrast with the flat and positively curved cases.

In this paper, we observe that the (spacelike) mean curvature flow of a submanifold in a (pseudo-)Euclidean space is equivalent to a harmonic-Ricci flow with coupling constant $\alpha=-1$ (or $+1$), for the corresponding Gauss map and the induced metric. The solitons of these two flows are also equivalent. As an application, we get a monotonicity formula for the spacelike mean curvature flow.

In this article, we investigate the global topology of a complete non-compact Riemannian $2$-manifold $\Sigma$ admitting a Schrödinger operator with non-negative potential and finite Morse index. While classical results of Fischer-Colbrie classify such manifolds under the assumption of vanishing index or geometric stability as immersed minimal surfaces in a Riemannian $3$-manifold, we show that the curvature at infinity $\lambda_\infty^*(\Sigma)$ of the Fischer-Colbrie metric $g^*$—a complete conformal metric determined by a positive function furnished by Fischer-Colbrie's theorem—governs the global topology and geometric rigidity of $\Sigma$ without imposing either assumption. More precisely, we derive a fundamental identity relating $\lambda_\infty^*(\Sigma)$ to the area growth of $(\Sigma,g^*)$, show that all critical points of the distance function $d_p^*$ from a fixed base point $p$ are confined to a bounded region, and, as a corollary, obtain a quantitative bound for the number of ends. We further distinguish two complementary geometric viewpoints. On the one hand, a quantitative condition on $\lambda_\infty^*(\Sigma)$ forces $\Sigma$ to be diffeomorphic to the Euclidean plane $\mathbb{R}^2$. On the other hand, when $\Sigma$ has exactly one end, another condition on $\lambda_\infty^*(\Sigma)$ guarantees that every Busemann function on $(\Sigma,g^*)$ is an exhaustion. By clarifying the relationship between these two regimes—the critical-point structure of distance functions relative to a base point and the global behavior of Busemann functions at infinity—we exhibit two complementary manifestations of how $\lambda_\infty^*(\Sigma)$ controls the global geometry and topology of $\Sigma$.

Let $(\mathfrak g,J,g)$ be a unimodular Hermitian Lie algebra containing an abelian ideal $\mathfrak a$ of real codimension two. We study the curvature behaviour of $g$ and show that, if $g$ has constant Chern holomorphic sectional curvature, then it must be Chern flat. We also show that, for every canonical metric connection $D_s^r$ of $g$ other than the Chern connection, if $D_s^r$ has constant holomorphic sectional curvature, then $g$ is Kähler flat, and in this case $\mathfrak g/\mathfrak a$ is abelian.

We develop a diffeological framework for the geometry of Euler–Reynolds subsolutions of the incompressible Euler equations. Passing to a lifted formulation in which the velocity, quadratic flux, and trace-free Reynolds stress are treated as independent variables, we construct a diffeological limit space obtained as the weak closure of smooth strict subsolutions. Its internal tangent spaces provide an intrinsic notion of infinitesimal deformation despite the absence of any underlying manifold structure. We prove that ambient realizations of internal tangent vectors satisfy the linearized Euler–Reynolds equations in the sense of distributions, thereby establishing a natural first-order theory for lifted weak limit spaces. We further describe the tangent directions compatible with the Euler locus and identify the kernels of the natural velocity, flux, and full-state observables. These results distinguish observable perturbations from hidden stress-gauge directions that encode infinitesimal variations of the Reynolds stress. Finally, we introduce a finite-mixture model for lifted Euler–Reynolds states whose differential realizes explicit tangent directions and relates Reynolds stress to infinitesimal phase splitting. Under a genericity assumption, every deviatoric stress tensor is realized by such a first-order mixture defect, yielding phase-counting bounds and a minimality result for the observable hierarchy.

We prove the existence of stability optimizers for the Yamabe inequality on closed Riemannian manifolds of dimension at least three with positive Yamabe invariant that satisfy two threshold conditions. Remarkably, the compactness threshold we uncover is different from the special case of the round sphere treated previously by the second author. More precisely, it is given by sequences blowing up in one instead of two bubbles, reflecting the compactness of Yamabe minimizers in the non-spherical case. Using the classical asymptotic analysis of Aubin–Schoen test functions, we prove that the stability constant is strictly below the one-bubble threshold in dimension at least six and when the manifold is not locally conformally flat. In the complementary case, namely in dimensions three through five or when the manifold is locally conformally flat, we find a new positive-mass-type condition which is sufficient for the strict inequality.

We investigate a class of Ricci-flat Kähler metrics on generalized conifolds constructed via gauged linear sigma models (GLSMs) with indefinite signature. By introducing shadow coordinates (superfields) entering the sigma model with negative signature kinetic term, we show that these GLSMs yield explicit Ricci-flat metrics on complex cones over products of projective spaces. We provide a general formula for the resulting Kähler potentials, along with detailed examples. Our results suggest new directions for the study of Calabi-Yau metrics and toric geometry, and raise interesting questions about the geometric meaning of indefinite signature models. We also give an interpretation in terms of a novel generalized Kähler gauging.

We prove existence and non-existence results for $L^2$ eigenforms for the Hodge Laplacian on complete non-compact Ricci flat manifolds with special holonomy, with a particular focus on gravitational instantons. We briefly describe the physical interpretation of these results in superstring and M-theory.

The Samuelson condition, a classical area-ratio condition for planar Lagrangian $2$-webs, is equivalent in local web coordinates $(u,v)$ to $\partial_u\partial_v\log|J_F(u,v)|=0$, where $F$ is the inverse web-coordinate map and $J_F$ is its Jacobian. For the tangent-line family $L_t:y=tx+h(t)$, let $\Psi(u,v)$, for $u\neq v$, be the intersection map of $L_u$ and $L_v$, write $J_\Psi$ for its Jacobian, and set $S_h(u,v):=\partial_u\partial_v\log|J_\Psi(u,v)|$. Near each diagonal point $(t_0,t_0)$ with $h''(t_0)\neq0$, the obstruction admits the decomposition $S_h(u,v)=1/(v-u)^2+R_h(u,v)$, where $R_h$ extends smoothly across the diagonal near $(t_0,t_0)$. We call $R_h(t,t)$ the Schwarzian residue and compute $R_h(t,t)=h^{(4)}(t)/(3h''(t))-(4/9)(h'''(t)/h''(t))^2=(1/2)\{\sigma,t\}=-\kappa_{\mathrm{aff}}(\sigma)(d\sigma/dt)^2$. Here $\{\sigma,t\}$ denotes the Schwarzian derivative of $\sigma$ with respect to $t$, where $\sigma$ is the equi-affine arclength parameter of the envelope $\gamma(t)=(-h'(t),h(t)-th'(t))$, and $\kappa_{\mathrm{aff}}$ denotes its equi-affine curvature, with the convention $\gamma_{\sigma\sigma\sigma}+\kappa_{\mathrm{aff}}\gamma_\sigma=0$. Thus the universal pole accounts for the local failure of the Samuelson condition, while the diagonal finite part carries affine-projective information about the envelope. We also derive the transformation law of the Schwarzian residue under reparametrization of the tangent-line parameter.

Let $(M,g)$ be a smooth orientable $2d$ Riemannian manifold of genus $\mathfrak{g}$ with Riemannian metric $g$ and connected boundary $\Gamma$. Let $\Lambda$ be the Dirichlet-to-Neumann map on $\Gamma$ and let ${\rm det}_\zeta(\Lambda)$ be its (modified, i. e. with zero mode excluded) $\zeta$-regularized determinant. It is well-known that the quantity ${\rm det}_\zeta(\Lambda)/|\Gamma|$ (where $|\Gamma|$ is the length of $\Gamma$) is a conformal invariant. It was shown by Edward and Wu (\cite{EV}) that this invariant equals one for $\mathfrak{g}=0$; in the case $\mathfrak{g}>0$ Guillarmou and Guillopé \cite{Guillarmou} found two explicit expressions for this invariant through the Rouelle and (respectively) the Selberg zeta-functions of the two surfaces of negative constant curvature from the conformal class of $(M,g)$: one is of infinite volume and complete whereas another has geodesic boundary. We present an elementary counterpart of the formulae of Guillarmou and Guillopé using the periods of holomorphic differentials on the double $2M$ of $M$ only. Our approach is based on the properties of the Hilbert transform of $M$ \cite{B,HilbKor} and the Kontsevich-Vishik-Friedlander-Guillemin regularization of the determinants of pseudodifferenial operators \cite{KV,Ww,F}. In particular, a connection between the length spectra of (uniformized) $M$, $2M$ and the periods of holomorphic differentials on $2M$ is established.

In Weinstein's symplectic "category", objects are symplectic manifolds and morphisms are canonical relations, which may not be composable; it is therefore only morally a category. We construct a true differential graded category which is morally dual to Weinstein's "category". The objects in this category are prequantum systems, associated to integral symplectic manifolds. The morphisms are a complex of differential forms twisted by a prequantum line bundle. We also consider the cohomology category, which turns out to be an (ordinary) linear category. In each case, Weinstein's morphisms are associated with currents dual to the forms. For the differential graded category, these are isotropic, or if preferred Lagrangian, currents, twisted by a section of the prequantum line bundle; in the case of cohomology, these currents are supported on integral Lagrangian submanifolds, equipped with a global covariant constant section of the prequantum line bundle. We then apply our methods to quantization, and show that for Kahler manifolds, a quantization in this category is related to holomorphic quantization. Along the way we show that in the case of prequantum systems, the Lefschetz action of ${\mathfrak {sl}}(2,\R)$ on differential forms, studied by Brylinski, Mathieu, Guillemin, and Tseng-Yau in the symplectic case, is promoted to an action of the superalgebra ${\mathfrak {osp}}(1|2)$ on twisted differential forms, with ${\mathfrak {sl}}(2,\R)$ as the even subalgebra. A first application of this superalgebra action is the vanishing theorem which shows the cohomology category is supported in middle dimension.

In 1970, Klingenberg proved a fundamental result on closed Riemannian manifolds with Anosov geodesic flows, asserting that such manifolds are without conjugate points. The proof, based on the Morse theory of the energy functional, had crucial gaps pointed out by Anosov, who saved the theorem by providing a completely independent proof. In this article, we fill the gaps in the original argument, and thus provide a full Morse theoretic proof of Klingenberg's theorem.

Let $(M^n,g)$ be a smooth compact Riemannian manifold of dimension $n\ge 5$. We show that the existence of a conformal metric with positive $Q$-curvature $Q_g$ and positive scalar curvature $R_g$ is equivalent to the positivity of both the Yamabe invariant $Y(M^n,[g])$ and the Paneitz operator $P_g$. For $n=5$, this equivalence confirms a conjecture of Gursky-Hang-Lin (2016, IMRN). Furthermore, assuming $Y(M^n,[g])>0$, $Q_g\ge 0$, and $Q_g\not\equiv 0$, we prove that both $R_g$ and $P_g$ are positive which resolves a problem of Hang-Yang (2016, CPAM). As a corollary, we show that the hypotheses of Gursky-Malchiodi (2015, JEMS) are equivalent to those of Hang-Yang (2016, CPAM).

A closed positively curved space curve is determined by its curvature and signed torsion up to an orientation-preserving rigid motion; that sign is the only place the ambient orientation enters. We ask what survives its loss, for closed embedded curves in $\mathbb R^3$ with $\kappa>0$ compared pointwise in a common arclength label. The answer is governed by the branch invariant $c(\tau)$, the number of components left by the infinite-order zero set of $\tau$: the smooth signed lifts of $|\tau|$ number exactly $2^{c(\tau)}$, and reduce to $\{\tau,-\tau\}$ precisely when $c(\tau)\le1$. Hence a given unsigned datum is carried by at most $2^{c(\tau)}$ classes modulo $SE(3)$, and by a single $E(3)$-orbit when $c(\tau)\le1$. Both extremes occur: for arbitrary knot types $K_1,\dots,K_m$ there is a datum with fibre exactly $2^m$ classes modulo $SE(3)$, realising all connected sums of the $K_i$ and their mirrors; under a chirality hypothesis these are $2^m$ knot types. Conversely, curves with only simple torsion zeros are open and dense, hence residual, among parametrised $C^r$ embeddings ($r\ge4$), and each is determined up to $E(3)$ by its datum. No uniform quantitative form of this rigidity exists; but on each stratum $\Delta=\inf_s\sqrt{\tau^2+(\tau')^2}\ge\delta>0$ with uniform $C^5$ and curvature bounds the orbit distance obeys a log-Lipschitz bound, whose optimal constants diverge as $\delta\downarrow0$ on the strata containing a fixed exact ambiguous pair. The engine is a one-dimensional inverse estimate for the signed square root, logarithmically optimal at that level.

Recently, Chen, Liu, and You (An extremal theorem for positive curvature of graphs, arXiv:2607.02297) proved an extremal theorem for positive Lin–Lu–Yau curvature. They further proposed the similar problems for other discrete curvature. In this paper, we prove a sharp extremal theorem for non-negative Bakry–Émery curvature with non-normalized Laplacian: every graph of order \(n\geq 7\) with more than \[ \binom {n}{2}-\lfloor{\frac{n}{2}\rfloor}-2 \] edges satisfies $CD(0,\infty)$, and this threshold is optimal.