Papers

Based on earlier work by Wiegerinck and Huang-Li, Hayashida-Kamimoto constructed unbounded non-smooth Reinhardt domains in \({\mathbb C}^n\), \(n\geq 3\), whose Bergman spaces are finite-dimensional and whose Bergman metrics have constant positive holomorphic sectional curvature \(2\). In this note, we prove a Sandwich Principle for the Hayashida-Kamimoto domains that allows us to construct, inside every prescribed sandwich, uncountably many pairwise biholomorphically inequivalent smooth non-Reinhardt domains whose Bergman metrics also have constant positive holomorphic sectional curvature \(2\). We also classify biholomorphisms between domains sandwiched by the domains of Hayashida-Kamimoto with the help of an affine rigidity theorem in the spirit of Cartan rigidity, which should be of independent interest. For explicit angular perturbation families, we determine the full automorphism group and obtain, in particular, an abundance of examples with trivial automorphism group.

Standing waves $e^{i\omega t}u(x)$ of nonlinear dispersive equations, such as the nonlinear Schrödinger and Klein–Gordon equations, have a profile $u$ that solves a nonlinear elliptic ODE of the form $\Delta u+f(u)=0$. We prove the existence of radial bound states with a prescribed number of nodal surfaces for this equation, on a large class of spherically symmetric manifolds and nonlinearities $f$. Under mild assumptions, we prove uniqueness of such bound states for the model nonlinearity $f(u)=|u|^{p-1}u-u$. Both existence and uniqueness results apply to some manifolds with a singularity at the origin.

We study the gauge of the travel time tomography problem for elliptically anisotropic planar media. Under this assumption the respective slowness surfaces consist of an inner ellipse and an outer circle. Hence, the singularities of seismic waves propagate on geodesics given by two Riemannian metrics of which the one related to shear waves is conformally Euclidean. We show that if both of these metrics are lens rigid then, up to boundary fixing diffeomorphisms, each elliptically anisotropic stiffness tensor is uniquely determined by the lens data whenever it has at least one isotropic point. If none of those points exists, then there are exactly two elliptically anisotropic stiffness tensors with the same lens data. Clearly, not all boundary fixing diffeomorphisms preserve the imposed structure of the slowness surface. We show that the set of well-behaved diffeomorphisms is never a subgroup of all boundary fixing diffeomorphisms. Our proofs are based on an interplay of algebra and geometry, and the key of the proofs is a characterization of the elliptic anisotropy via several polynomial conditions.

By establishing a weighted Lojasiewicz-Simon inequality for maps which are close to a fixed harmonic map modulo scaling, we prove Topping's conjecture on continuity of the body map in the case of finite time singularities at which only isolated bubbles form.

We obtain sharp isoperimetric-type upper bounds for Steklov eigenvalues on warped products whose bases are compact Riemannian manifolds with boundary and whose fibres are closed Riemannian manifolds. These bounds involve the volume of the warped product and of its boundary, as well as the eigenvalues of the Laplace operator on the fibre and the Steklov eigenvalues of the base. After separation of variables, these bounds are obtained by comparison with auxiliary Helmholtz-Steklov problems on the base involving weighted Laplace operators naturally induced by the warping function. We also study the effect of various natural geometric constraints, such as the warping function being identically equal to one on the boundary. The proofs make use of various spectral approximation methods, including homogenisation on manifolds when the fibres are one-dimensional.

In [1, Proposition 8.1], Alexakis and Mazzeo showed that the renormalized area of minimal surfaces in the hyperbolic space $\mathbb{H}^3$ approaching the ideal boundary $\partial_\infty\mathbb{H}^3$ is the negative of the Willmore energy of their Euclidean conformal compactifications. In this paper, we investigate whether there are other geometric surface equations for which these two quantities satisfy a universal proportionality relation. We answer this question in the affirmative by extending the Alexakis–Mazzeo result to a broader class of geometric equations containing, as two distinguished cases, the equations defining generalized singular minimal surfaces in $\mathbb{R}^3$ and linear Weingarten surfaces in $\mathbb{H}^3$. Moreover, for rotational linear Weingarten surfaces, we express both energies in terms of generalized hyperelliptic integrals and show that their doubles form a family of rotational tori in $\mathbb{R}^3$ whose Willmore energies realize all values in $(2\pi^2,\infty)$, with a particular subfamily converging to the Clifford torus in $\mathcal{W}^{2,p}$ for every $1\leq p<\infty$, as the optimal value $2\pi^2$ is approached.

We prove that on a compact locally conformally Kähler manifold, the existence of a non-trivial parallel vector field with respect to a Gauduchon connection $\nabla^t$, for $t\ne \frac{1}{3}$, forces the metric to be either Vaisman (for $t=-1$), globally conformally Kähler (for $t=1$), or Kähler for all other values of $t$. As a consequence, we characterize compact Vaisman manifolds in terms of their Bismut holonomy group. Moreover, we show that in complex dimension two, the existence of a non-trivial $\nabla^\frac{1}{3}$-parallel vector field forces the metric to be globally conformally Kähler.

By a celebrated result of Birman and Series, the set of simple complete geodesics form a nowhere dense subset of a hyperbolic surface. We show that a segment of finite length is in a (small) neighborhood of this set if it can be "extended simply" in both directions for long enough. We deduce that, for large enough genus $g$, there exist surfaces with a Birman-Series set that is roughly $\frac{1}{g^{\sfrac{2}{3}}}$-dense.

On the Riemannian Schwarzschild–Tangherlini metric of dimension $n\geq 4$, we show that the destabilizing TT tensor of Ozuch lies in the image of the “TT-tensor machine”. By continuity, it can then be approximated, in suitable weighted Sobolev spaces, by smooth compactly supported TT tensors, on which the stability quadratic form is still negative. This proves instability in the usual sense for noncompact manifolds. To our knowledge, no exact smooth compactly supported destabilizing TT tensor was known before, even in 4D.

We produce the first examples of singular isoperimetric boundaries with non-zero constant mean curvature in Riemannian manifolds. The main technical ingredient is establishing that, for constant mean curvature hypersurfaces containing isolated singularities modelled on strictly stable, strictly minimising regular tangent cones, weak strict stability implies local unique volume-constrained minimisation.

We study the recovery of a three-dimensional Riemannian manifold from the areas of minimal surfaces. We show that areas of minimal surfaces determine the metric on manifolds foliated by minimal surfaces, up to boundary-fixing isometry. This resolves the conjecture on boundary rigidity for least-area data in an exterior formulation, even without topological restrictions on the minimal surfaces. We also prove global rigidity for Einstein metrics on compact three-manifolds with boundary, assuming $\(\pi_1(M,\partial M)=0\)$ and a non-trapping condition. The Einstein result includes the

We study the Yang–Mills flow on $\mathbb{S}^4$ restricted to $\mathrm{SO}(4)$-equivariant connections on the trivial $\mathrm{SU}(2)$ bundle. It was expected that antipodal instanton–anti-instanton configurations would lead to infinite-time singularities along the flow. We show that this does not occur. Moreover, for any smooth equivariant initial condition, the flow converges to the trivial flat connection.

We classify all nonconstant smooth harmonic maps $\mathbb{S}^2\to \mathbb{CP}^n$ and $\mathbb{S}^3\to \mathbb{CP}^n$ whose 0-homogeneous extensions are minimizing: they are, respectively, isometric embeddings or holomorphic isometric embeddings composed with a Hopf map.

We construct two word metrics on the same seven-dimensional torsion-free nilpotent lattice whose asymptotic cones are isometric, but for which every quasi-isometry has multiplicative constant at least $1+\varepsilon$, for one fixed $\varepsilon>0$. We use a convex invariant of a fixed quasi-isometry $\Psi:N\to M$. The displacement cocycle of $\Psi$ and the right-invariant means on $N$ determine a compact convex set $\mathcal A(\Psi)\subseteq\operatorname{Hom}(V_N,V_M)$. If $\mathscr G_{\mathrm{pt}}(\Psi)$ denotes the first-layer maps of the graded homomorphisms that occur as blow-downs of $\Psi$ with varying basepoints, then $$ \mathcal A(\Psi)=\operatorname{conv}\mathscr G_{\mathrm{pt}}(\Psi), \qquad\operatorname{Ext}\mathcal A(\Psi)\subseteq\mathscr G_{\mathrm{pt}}(\Psi). $$ For every extreme point $L\in\mathcal A(\Psi)$, the dual map $L^*$ is also the degree-one part of an isomorphism of the full real cohomology rings. This cohomological compatibility, together with the realization of first-layer Pansu differentials by invariant means, yields the multiplicative constant gap stated above.

Let $X$ be an irreducible reduced compact complex space of dimension $n$, and let $\omega$ be a Hermitian metric on $X$. For each $1\le k\le n-1$, we prove that there exist a constant $c_k$ and a smooth positive function $\rho$ on $X_{\mathrm{reg}}$ such that $\ii\partial\bar\partial(\rho\omega^k)\wedge\omega^{n-k-1}=c_k\rho\omega^n$. The function $\rho$ is bounded above and bounded away from zero. The constant $c_k$ is unique, and $\rho$ is unique up to a positive multiple among bounded smooth positive solutions. We also prove that the full sign set $\{-1,0,1\}$ of the generalized Gauduchon constants $c_k$ on a compact complex manifold is preserved by finite sequences of blow-ups with smooth centers.

We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial $Q$-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao–Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.

We prove a sharp Zhong-Yang type lower bound for the first nonzero eigenvalue of the $p$-Laplacian on closed manifolds with small integral Ricci curvature for $p \geq 2$. The proof combines a global elliptic estimate obtained by Sobolev truncation with strict perturbations of one-dimensional comparison profiles.

We construct a geometric theory on the brane space $S^2M=C^\infty(S^2,M)$ associated to a gerbe with connection on $M$. Normalized logarithmic holonomy around the latitude circles defines an average holonomy line bundle on the brane space, whose connection and averaged flux give an equivariantly flat superconnection. We prove that the completed periodic exotic twisted cohomology defined by this superconnection localizes to the twisted cohomology of $M$. This brane space localization happens in two stages: circle localization lands on the free path space of $M$, and contracting the paths, after a correction of the twisting form, brings us back to $M$. Parallel transport along the meridians lets us average gerbe module connections, after which the circular iterated-integral construction applies. The resulting twisted brane Bismut-Chern forms restrict to the twisted Chern forms on $M$; their variation is described by twisted brane Bismut-Chern-Simons forms. For a T-dual pair, transporting the classical Hori morphisms through localization yields graded genus-0 T-duality on brane cohomology. The composite of each morphism with its dual is minus the Euler operator in the level variable, so these morphisms are isomorphisms at every nonzero level. At level one these morphisms are compatible with the brane character and twisted $K$-theory T-duality.

We implement (in a concrete example) a procedure of Dubrovin which gives a braid group action on spaces of solutions of ordinary differential equations such as the quantum differential equation of complex projective space. Such actions originated in physics for the tt* (topological-antitopological fusion) equations. We make the procedure systematic by utilizing an algebraic description of braid group actions on distinguished bases. We also generalize to the case of (arbitrary) solutions of the tt*-Toda equations, one of which corresponds to the quantum cohomology of complex projective space.

We construct compact, embedded solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ that arise from various non-compact initial data and exist for a finite time.

We construct closed Levi non-degenerate CR manifolds with non-vanishing Chern monomials. For every $n \geq 2$ and $\lceil n / 4 \rceil \leq m \leq \lfloor n / 2 \rfloor$, there exists a $(2 n + 1)$-dimensional example of signature $(\lfloor n / 2 \rfloor, \lceil n / 2 \rceil)$ on which all Chern monomials with positive indices at most $m$ and total weight at most $n - m$ are non-zero in cohomology with real coefficients. Taking $m = \lceil n / 4 \rceil$ gives examples with $c_{1}^{\lfloor 3 n / 4 \rfloor} \neq 0$. For $n = 4$ or $n \geq 6$, the latter examples show that the Chern-class vanishing theorem for strictly pseudoconvex CR manifolds does not extend to the Levi non-degenerate case. In these dimensions, their underlying manifolds admit no strictly pseudoconvex CR structures. The construction uses circle bundles over products of integral pseudo-Kähler manifolds.

Automatic differentiation (AD) is a cornerstone of modern machine learning. Since all AD frameworks implicitly fix a basis of Euclidean space to define gradients and Jacobians, they natively process mathematical objects that are arrays of a fixed set of coordinates. This makes it awkward to express objects such as rotations, volume forms, and curvatures that arise, for example, in applications of AI to science. Bespoke workarounds lack reusability. We show that (i) a single, basis-independent derivative operator can represent many geometric differential operations, and (ii) this operator can be expressed entirely in terms of existing AD primitives. This reusable transformation can be composed with JAX batching, compilation, and hardware acceleration. Geometry enters as algebraic data. We experimentally demonstrate its reuse across different manifold operators, its composition through a spacetime inverse-problem pipeline, and its use in optimization over different rotor subgroups.

We show that for a generic compact initial surface in $\mathbb{R}^3$, and any generic compact initial mean convex hypersurface in higher dimensions, the mean curvature flow encounters only finitely many singularities. This resolves the full generic Morse regularity conjectures of Ilmanen and Colding-Minicozzi-Pedersen in the above two cases. More generally, we show that if a mean curvature flow of a compact hypersurface has only multiplicity one spherical and cylindrical singularities, then after a small perturbation of the initial condition only non-degenerate cylindrical and spherical singularities appear.

We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot split into several nearby eigenvalues, while an isotypic limiting eigenspace imposes corresponding divisibility constraints on the multiplicities in the approximating cluster. We then construct a one-parameter equivariant Steklov homogenisation on closed orientable surfaces, allowing Laplace spectral representations to be transferred to Steklov spectral clusters. Applied to the hyperbolic surfaces of Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault, this yields first Steklov multiplicities 16 and 21 in genera 10 and 17, and at least 24 in genus 37. These examples give counterexamples to Jammes's conjecture on first Steklov multiplicity.

We show how triality, the symmetry permuting the three 8-dimensional representations of $\mathrm{Spin}(8)$, can also appear as a symmetry of geometric objects such as connections and Dirac operators. Take any Riemannian metric $g$ on the manifold $M = \mathrm{SU}(3)$ and let $\nabla$ be a metric connection on $TM$. Using an explicit model of Clifford multiplication by deformed Okubo products, we combine spinorial lifts with isometries $TM \simeq S^+ \simeq S^-$ to show that $\nabla$ can be rotated into two new metric connections on $TM$. So there is a canonical $\mathbb{Z}_3$-orbit of metric connections determined by $(g, \nabla)$, with no other choices. The three connections coincide when $g$ is bi-invariant and $\nabla$ belongs to the Cartan-Schouten line, but in general they are different. The rotations of $\nabla$ preserve the norm of the curvature, so they define a new $\mathbb{Z}_3$-symmetry of the moduli space of Yang-Mills connections on $TM$. They also preserve the Chern-Simons 3-form and adjoint Yang-Mills-Higgs functionals. After choosing an orientation on $M$, the rotated connections define three self-adjoint, Dirac-type operators on the spinor bundle. We prove that these operators are natural, in the sense that they commute with the Kosmann derivatives of spinors along all invariant Killing vector fields that preserve $\nabla$. We also show that they are not isospectral, in general. This is verified by computing the simplest eigenvalue for the Levi-Civita connection and observing that it splits among the three operators as the bi-invariant metric on $M$ is deformed to one with isometry group $(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)) / \mathbb{Z}_6$. Finally, we outline how the constructions in the paper may extend to general 8-manifolds with a $\mathrm{PSU}(3)$-structure.

We construct a polar action of $\mathrm{SU}(3)$ on a closed simply connected $11$-manifold with a compact three-dimensional section that is not injectively immersed. The section is injective over the regular part and has two distinct local branches along a singular stratum. All isotropy groups are connected, and all orbits are simply connected. This gives a negative answer to the injectivity question raised by Grove and Ziller.

We present a generalization of T-duality, called $T_2$-duality, that captures certain aspects of U-duality in M-theory. In particular, it features backgrounds that contain a 2-gerbe over a principal torus bundle, structures familiar from eleven-dimensional supergravity backgrounds. In special cases, it also incorporates S-duality in a precise sense. Our main result is a theorem characterizing $T_2$-dual pairs through spans of categorified principal bundles with connections, giving explicit rules for constructing a $T_2$-dual for a given background. A number of explicit examples illustrate how the various characteristic classes are mapped between the left and right backgrounds. For two-dimensional torus fibers, we also relate our construction to the combination of dimensional reduction and T-duality that links eleven-dimensional supergravity to type IIB supergravity. Finally, we discuss how $T_2$-duality arises from a Buscher-like span of 3d sigma models.

Let $M$ be a closed manifold of arbitrary dimension. The length spectrum of a negatively curved metric $g$ on $M$ is the set of lengths of all possible closed geodesics of $g$. We prove that metrics whose length spectrum is exponentially sparse, i.e., the gaps between distinct lengths are bounded below by a quantity decaying exponentially in the length, are prevalent, and hence dense, among $C^k$-smooth negatively curved metrics on $M$ for sufficiently large $k$. Moreover, unlike in all known constructions, the exponent in our bound grows only sublinearly in the regularity $k$. This work is motivated by our recent result on prevalent unmarked length spectral rigidity for expanding circle maps and a very recent result by DeWitt, Durham, Reber, and O'Hare that $C^k$-smooth negatively curved metrics with exponentially sparse spectra are locally rigid.

Let $n \geq 3$. We prove that if $j \colon (\Sigma^n,g) \to (\mathbb{R}^N,dx^2)$ is an isometric immersion, then \begin{equation*} \int_\Sigma \left( \lvert \nabla u \rvert^2 + \frac{n(n-2)}{4}\lvert H \rvert^2 u^2 \right) \geq \frac{n(n-2)}{4}\mathrm{Vol}(S^n)^{2/n}\left( \int_\Sigma \lvert u \rvert^{\frac{2n}{n-2}} \right)^{\frac{n-2}{n}} \end{equation*} for all $u \in W^{1,2}(\Sigma)$, and characterize immersions for which equality is realized. Our proof exploits the conformal invariance of the inequality, and hence extends to sharp Michael–Simon inequalities for submanifolds of the sphere, of hyperbolic space, and of other conformally developable manifolds. As an application, we derive Chen–Fenchel–Willmore-type inequalities relating $L^p$-norms of the mean curvature vector to the volume of a compact submanifold of a simply-connected spaceform.

We study strict inverse systems of locally convex Lie groupoids and Lie algebroids. We show that strict inverse limits of Lie algebroids and Lie groupoids inherit natural Lie structures and establish a natural isomorphism of functors $\varprojlim\circ\operatorname{Lie}\cong\operatorname{Lie}\circ\varprojlim$. We then consider inverse systems of integrable Banach-Lie algebroids and their source-simply connected levelwise integrations. We give conditions under which these integrations form a strict inverse system of Lie groupoids and hence their inverse limit integrates the inverse-limit algebroid. Under additional hypotheses on the source-fiber inverse sequence, we describe the obstruction to source-simply connectedness of the resulting integration in terms of the derived inverse limit $\varprojlim^{1}\pi_2$ of the second homotopy groups of the source fibers. We illustrate the theory with several examples: inverse limits of jet groupoids, current groupoids, and gauge groupoids, including the corresponding inverse-limit Atiyah algebroids and a source-simply connected inverse-limit integration arising from the quaternionic Hopf bundle.