Papers

We show that for an embedded two-sided minimal hypersurface in $\mathbb{R}^N$, there is a lower bound for the index in terms of the first Betti number and the number of ends, using ideas from the recent work of Chodosh–Gianocca. This estimate is sharp for the higher-dimensional catenoid. We also obtain a $\operatorname{Spin}(7)$ analogue of Theorem 10.1 of Chodosh–Gianocca: a complete two-sided minimal immersion $M^7\to\mathbb{R}^8$ of index one and finite total curvature is a higher-dimensional catenoid.

We study the singularity type and models of the Kähler–Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler–Einstein manifolds $N := N_1 \times \cdots \times N_r$, with metric constructed using the ansatz considered in \cite{DW2011}, \cite{WW} et. al. In the earlier work \cite{FT} by the authors, we considered the “two-bolt” case where both ends of the foliation close with the “bolt” $N$. In this article, we continue our work on the more subtle “nut-bolt” and “two-nut” cases. The former has one end of the interval closes with a nut-type collapse (i.e. $N' := N_2 \times \cdots \times N_r$) and the other with a bolt (i.e. $N$). The compactification $\widehat{M}$ is then a $\mathbb{CP}^{m+1}$-bundle over $N'$. The “two-nut” case is one that both ends close with nut-type collapses, necessarily two of the $N_i$'s must be $\mathbb{CP}^{m_0}$ and $\mathbb{CP}^{m_\ell}$, and the compactification $\widehat{M}$ is a $\mathbb{CP}^{m_0+m_\ell+1}$-bundle over $\prod_{k\geq 3}N_k$. We proved that in all “two-bolt”, “nut-bolt” and “two-nut” caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be $(\Sigma^{m+1}, g_\Sigma(t)) \times (\mathbb{C}^{k}, \textrm{flat})$ with $m, k \geq 0$, where $\Sigma$ is one of the following: $\mathbb{CP}^{m+1}$, $\textrm{Tot}(\mathcal{L}^{\oplus(m+1)})$, or a projectivization $\mathbb{P}\big(\mathcal{O}^{\oplus(m_0+1)} \oplus \mathcal{L}^{\oplus(m_\ell+1)}\big)$ with $m_0 + m_\ell = m$, and $\mathcal{L}$ is a line bundle over the product of some of the $N_1, \cdots, N_r$ factors. The metric $g_\Sigma(t)$ is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.

We establish two nonuniqueness phenomena on $\mathbb{CP}^1$. First, distinct normalized positive-mass model potentials can have the same non-pluripolar Monge–Ampère measure. By the contact-set formula, their zero-contact sets agree $\omega_{\mathrm{FS}}$-almost everywhere. Moreover, both have zero Lelong number at every point and identical multiplier ideal sheaves at every positive scale. Second, we construct distinct maximal test curves whose zero-contact sets agree pointwise at every parameter. Their inverse Legendre transforms are distinct bounded weak geodesic rays from zero with the same pointwise right initial tangent.

Let $(M,g,J)$ be a closed Kähler manifold satisfying $\operatorname{Ric}\geqslant g$. We establish improved Liouville theorems for the Euler–Lagrange equations associated with the Beckner–Sobolev inequalities by incorporating the first positive eigenvalue of the $\bar\partial$-Laplacian into a differential-identity argument. As a consequence, we obtain improved Sobolev and Beckner inequalities that refine the known Riemannian and Kähler estimates when the first eigenvalue is sufficiently large. We also derive new upper bounds for the diameter of $(M,g)$.

The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in $\mathbb{C}^2$. We modify a standard construction of nonembeddable smooth strongly pseudoconvex CR 3-manifolds so that the condition of having a nowhere-zero closed section of the canonical bundle is preserved, thus providing a strongly pseudoconvex counterexample to the Trautman conjecture.

Symmetry reductions of partial differential equations (PDEs) inherit more geometric structures than other types of reductions: invariant conservation laws, variational structures, and, under suitable conditions, Hamiltonian-type structures of the original model descend to the reduced model through the mechanism of invariant reduction. This paper develops computational algorithms that carry out such reductions explicitly. We use the interpretation of variational $p$-forms as conservation laws of an enlarged system — the (degree-shifted) tangent system — consisting of the original equations together with their linearizations, in which the perturbation variables are treated as anticommuting. This allows us to formulate the algorithms using well-known concepts from the theory of conservation laws, naturally adapted to the graded-commutative setting. The reduction of conservation laws, variational $1$-forms, and presymplectic structures thereby becomes a single algorithmic procedure. We present (i) a homotopy-based reduction algorithm for systems of evolution equations, implemented in Maple; (ii) a descent reduction algorithm applicable to general $\ell$-normal systems for $p>0$; and (iii) a simple reduction algorithm available for point symmetries under suitable conditions, including the description of reductions in terms of systems involving fewer independent variables. Examples include nonlinear evolution equations in one and two spatial dimensions, the Laplace equation, the incompressible Euler equations, and the cotangent system of Pavlov's equation.

We prove uniqueness of tangent flows for mean curvature flow with free boundary at singularities modeled on half-cylindrical self-shrinkers. More precisely, if a tangent flow is of the form $\mathbb{S}^{n-k} \times \mathbb{R}^k_+$ or $\mathbb{S}^{n-k}_+ \times \mathbb{R}^k $, then it is unique. This provides the free-boundary analogue of the uniqueness theory for cylindrical singularities.

Thirty years ago in “On the Maslov Index” the present authors with their late collaborator Prof. Ronnie Lee of Yale University presented an axiomatization of the Maslov index, an integer-valued invariant of continuous, piecewise smooth paths of pairs of Lagrangians. Four distinct methods of definition of such Maslov indicies were given; each satisfying the axioms. Here two more are offered and comparisons are made to the recent work of Her and Zhong defining an invariant of a path of symplectic matrices motivated by earlier work of Lidskii and Gelfand, Salomon and Zehnder.

This expository article is based in part on the author's lecture series, "Einstein Metrics, Four-Manifolds, and Differential Topology," which was delivered as a 2026 Cours de la Chaire d'Excellence for the Fondation Sciences Mathématiques de Paris.

We study collapsing finite-time singularities of the unnormalized Kähler–Ricci flow on Fano bundles arising in the analytic minimal model program. For a Fano bundle $X^n\rightarrow Y^m$, we prove a maximal splitting theorem by the Kähler-Ricci flow that every tangent space at a fixed limiting point splits globally as \((\C^m,g_{\rm E},J_0)\times(Z',d',J')\). If the fibre has complex dimension one, we prove a global Type-I bound for the full curvature tensor and show that the ambient and intrinsic diameters of every fibre are uniformly comparable to \(\sqrt{T-t}\). Furthermore, every tangent flow at any fixed limiting point is the round shrinking cylinder \(\C^m\times\PP^1\).

This article equips the universal moduli spaces of non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic spinors and eigenspinors with tame Fréchet manifold structures and proves that their natural projections to the corresponding parameter spaces are uniformly Fredholm. A unified construction yields the deformation theories of harmonic spinors and eigenspinors in dimension three and of chiral harmonic spinors in dimension four, recovering earlier work of Donaldson, Parker, and Takahashi. As an application, a deformation theory for non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic self-dual $2$-forms in dimension four is established.

We prove a sharp comparison, with Obata-type rigidity, for all Neumann eigenvalues of one-dimensional $\mathrm{CD}(1,2)$ spaces against the Legendre model, under a convexity condition on the density. It follows that minimizers of the $k$-th Laplace eigenvalue among closed surfaces of Gaussian curvature at least $1$ cannot collapse in the measured Gromov-Hausdorff completion, for every $k\ge2$. We also give a variational proof that smooth minimizers are round.

We prove that the standard cohomogeneity-one action of \(SU(3)\) on \(S^7\), with principal orbit the Wallach flag manifold \(SU(3)/T^2\), admits infinitely many invariant Einstein metrics. The proof uses a detection approach to the Einstein boundary-value problem. Starting from one singular orbit, we follow the Einstein solutions only to a canonical hypersurface and measure there how far they are from closing smoothly. When the singular-orbit scale becomes small, this closing problem is governed by a Ricci-flat limiting solution. The linearisattion about its limiting cone has an oscillatory mode. As the scale shrinks, this oscillation repeatedly changes the sign of the closing error, producing infinitely many parameter values for which the metric closes. The closing scales satisfy an asymptotic logarithmic quantization law, with successive ratio tending to \(e^{-2\pi/\sqrt{15}}\). We also determine the geometry of the resulting sequence. Away from the two singular orbits the metrics converge to the singular sine cone over the non-normal Einstein metric on \(SU(3)/T^2\), while after rescaling by the square of the singular-orbit scale at either end they converge to the same complete Ricci-flat threshold metric. Their curvature is of order \(b_n^{-2}\), so the logarithmic phase law induces a corresponding quantization of the focal curvature scale.

We consider on a Riemannian manifold $M$ the Leibenson equation \begin{equation*}\label{eqabs}\partial _{t}u=\Delta _{p}u^{q},\end{equation*} where $p>1$ and $q>0$. When $q(p-1)\geq 1$, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound $V(x_0, r)\leq \exp\left(C r^{\frac{p}{p-1}}\right)$ for some $x_{0}\in M$ and all large enough $r>0$. When $q(p-1)< 1$, we prove this property assuming $V(x_0, r)\leq Cr^{N}$ and $p>N[1-q(p-1)]$, which matches the threshold in $\mathbb{R}^{n}$ with $N=n$. We also show that solutions on the hyperbolic space $\mathbb{H}^{n}$ have a finite extinction time in the case $q(p-1)< 1$, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case $q(p-1)=1$, we also prove a $L^{p-1}$- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.

Let $(M,\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $\mu$. We study the small-scale limit of the metric-ball mean-value operator \[ A_hf(x)=\frac{1}{h^{2}\mu(B(x,h)) }\int_{B(x,h)}(f(q)-f(x))\, d\mu(q). \] Exact homogeneity yields the pointwise limit on Carnot groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $\mu$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.

Motivated by the problem of finding obstructions to the existence of symplectic structures in dimensions higher than four, we introduce an elliptic system of equations on almost Hermitian manifolds that reduces to the Seiberg–Witten equations with Taubes' perturbation in dimension four. To define the equations, one needs to choose a spin$^{\mathbb C}$ structure on the manifold. We prove that the system has index zero for the canonical spin$^{\mathbb C}$ structure. Moreover, on almost Kähler (symplectic) manifolds, we construct a canonical solution and prove its transversality with large values of the perturbation parameter, while in dimension six we obtain uniqueness of the canonical solution under a certain assumption. We also show that on Kähler manifolds the equations reduce, under a natural ansatz, to the vortex equations. These results provide evidence toward a possible higher dimensional gauge theory capable of producing invariants of almost complex structures and ultimately, obstructions to compatible symplectic forms.

In this paper, we derive some new relative volume comparison theorems and Bishop-Gromov volume comparisons on Finsler metric measure manifolds, all of which are controlled by the integral weighted Ricci curvature. In particular, we establish a Bishop-Gromov volume comparison theorem for nonconcentric balls. Based on these, we prove a theorem of Bonnet-Myers type on Finsler metric measure manifolds with integral weighted Ricci curvature bounds.

We resolve the long-standing problem of establishing interior \(C^2\) estimates for admissible solutions of the graphical scalar curvature equation in every dimension \(n\ge 3\). More precisely, we prove interior curvature estimates for admissible solutions to the constant graphical scalar curvature equation. The proof combines Jacobi inequalities with a two-surface maximum principle and a two-surface Pogorelov estimate.

In this paper, we study the torsion problem on bounded, smooth, strictly horo-convex domains in hyperbolic space. We prove that if the diameter of the domain is sufficiently small, then $-\sqrt{u}$ is strictly convex, where $u$ denotes the torsion function. The main ingredients of our proof are the constant rank theorem, a boundary convexity estimate, and a deformation argument for the domain. We also show that the exponent $1/2$ is optimal, even among strictly horo-convex domains of arbitrarily small diameter.

Let $(M^n,g)$ be complete, let $u>0$, and assume $\operatorname{Ric}_g-\alpha\frac{\Delta_g u}{u}g\ge (n-1)\kappa g$. We introduce mixed radial balls associated with the conformal metric $u^{2\alpha}g$. For these mixed radial balls, we obtain a space-form-sharp model comparison and polynomial weighted volume growth without pointwise bounds on $u$. As applications, we give a radial derivation of the spectral Bonnet–Myers and sharp volume theorem of Antonelli–Xu, and a short volume growth derivation of the stable Bernstein theorem in $\mathbb{R}^4$.

We study formal deformations of the natural $\osp(n|2)$-action, $n\geq 3$, on the superspace $\Sc^n_d=\bigoplus_{k\geq 0}\Fc^n_{d-\frac{k}{2}}$ of symbols of linear differential operators on weighted densities over $\R^{1|n}$. Starting from the first cohomology space computed in \cite{10}, we compute the cup-product $\Hd^1\vee \Hd^1\to \Hd^2$ which carries the quadratic obstructions. The answer is governed by the $\osp(n|2)$-invariant operators $A_k=\eta_1\cdots\eta_n\partial_x^{k-1}$: the two cocycles $h_k$ and $\widetilde{h}_k$ spanning the off-diagonal part of $\Hd^1$ are exactly the two derivatives of the coboundary of $A_k$ with respect to the two weights. Consequently, all the products of two off-diagonal classes and all the products of two diagonal classes vanish, and the whole obstruction is carried, for each $k$, by a single non-trivial 2-cocycle $\Omega_k$. If $2d\notin\N$ the space $\Hd^1\vee\Hd^1$ is identically zero, so every infinitesimal deformation is integrable. If $2d=m\in\N$ we obtain exactly $m$ quadratic integrability conditions, $\tau_{2-n-k}(t_k-\widetilde{t}_k)+\tau_k\widetilde{t}_k=0$, $1\leq k\leq m$, and we prove that they are also sufficient: no condition of order $\geq 3$ occurs and the versal deformation is of degree one in the parameters. In particular every integrable formal deformation is equivalent to its infinitesimal part.

We study locally conformally flat (LCF) Riemannian manifolds with nonnegative or positive scalar curvature (PSC), using the conformal boundary of the developing image. For closed oriented LCF $n$-manifolds with PSC and infinite fundamental group, $n\ge5$, we bound the macroscopic dimension of their Riemannian universal covers by $\lfloor(n-1)/2\rfloor$, establish the existence of a nontrivial homotopy group above the middle dimension, and, under mild additional hypotheses, bound the Hausdorff dimension of the limit sets of their Kleinian groups in the interval $(1,(n-2)/2)$. In particular, no closed aspherical manifold admits an LCF metric with PSC. If the scalar curvature is at least $n(n-1)$ and the manifold is not isometric to the round sphere, then every smooth nonzero-degree map to the sphere expands somewhere when its Kleinian group is elementary, while the developing map expands somewhere whenever the fundamental group is infinite. We prove that the space of LCF metrics with PSC and its moduli space are contractible in dimension three for finite fundamental group and for $S^2\times S^1$, and that the moduli space is empty or contractible for smooth manifolds homeomorphic to spherical space forms but not diffeomorphic to $S^n$ in dimensions $n\ge4$. For complete open simply connected LCF manifolds of nonnegative scalar curvature, we obtain Euclidean rigidity under additional topological hypotheses at infinity (and, in dimension three, from vanishing second homology alone). We also show that, in dimensions $n\ge4$, the Euclidean conclusion can fail when neither of the two additional topological hypotheses is assumed: we construct complete contractible examples of PSC that are not homeomorphic to $\mathbb{R}^n$.

We compute the $\log p$-part of the asymptotic expansion of the holomorphic torsion of the $p$-th power of a positive line bundle in terms of characteristic classes. This is done for orbifold and equivariant torsions. In the equivariant case the exponents in the expansion are less than or equal to the dimension of the fixed point set. We deduce a corresponding equivariant arithmetic Hilbert-Samuel theorem in Arakelov geometry.

We derive optimized upper bounds for the first Dirichlet eigenvalue of a bounded domain under lower bounds for the conformal Ricci tensor of Shen and Ye. The method also yields sharp estimates for the bottom spectrum on complete noncompact manifolds, a four-dimensional application involving $Q$-curvature, and local and global spectral-Ricci extensions of Cheng's estimate. As a consequence, we obtain a sharp comparison between the spectra of the Laplacian and the conformal Laplacian in terms of the smallest Schouten eigenvalue.

We prove existence of closed non-planar $p$-elasticae for general exponents $p\in (1,\infty)$. In particular, we show that for any $p \in (1,\infty)$, there exists a countable family of non-planar $p$-elasticae, which are realized as torus knots. This generalizes well-known results of Langer–Singer for the quadratic case $p=2$ to general exponents $p\in (1,\infty)$.

We construct the first examples of isospectral but not strongly isospectral spherical space forms with non-cyclic fundamental groups. We obtain infinitely many such pairs with isomorphic fundamental groups of Type I.

Let $S$ be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of $S$, unless $S$ is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the $n$-dimensional case, $n>2$, where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when $S$ is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.

In this note, we prove the existence of smooth complete admissible hypersurfaces in hyperbolic space with constant higher order mean curvature and a prescribed asymptotic boundary at infinity. The result holds for a large part of indices in the subcritical range where the concavity inequality loses its effectiveness. We supply with new arguments to reduce the proof to a semi-convex situation in which case the concavity inequality can play a role.

We prove stability of the Riemannian positive mass theorem in all dimensions, extending the Dong-Song stability theorem. For a sequence of complete asymptotically flat manifolds with nonnegative scalar curvature and ADM masses tending to zero, excising domains whose boundary areas tend to zero yields exterior regions converging to Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof constructs global coordinates from minimal graphs and controls their Hessians using scalar solutions of the conformal Laplace equation on the associated graph metrics.

In this paper we establish removable singularities results for Einstein metrics and for metrics with Ricci curvature bounded below. Let $n\geq 2$. On a closed $n$-manifold, we show that an $L^\infty$-Riemannian metric whose Ricci curvature is bounded below outside a singular set of codimension $> 3- \frac{1}{n-1}$ canonically extends to an $\mathrm{RCD}$ space. As a consequence, using a new removable singularity theorem for Einstein metrics, we prove that in dimension $4$ any Einstein metric with $L^\infty$ singularities of codimension $>3-\frac{1}{3}$ extends smoothly across the singular set, possibly after changing the smooth structure. In higher dimensions, we construct a $C^{1,\alpha}$-Riemannian manifold structure on the regular set of a non-collapsed $\mathrm{RCD}$ space that is a Riemannian manifold with bounded $|\mathrm{Ric}|$ outside a set of codimension $>2$. Our results can be used to give a proof of Schoen's conjecture on scalar curvature singularities for metrics that are either continuous, or $L^\infty$ and sufficiently close to a smooth background metric.