Papers

We study the Jacobi spectrum and low-index rigidity of complex Hopf fibrations. We first give a representation-theoretic description of the Jacobi operator of a homogeneous Riemannian fibration and apply it to \[ \pi:(\mathbb{S}^{2n+1},g_\epsilon)\longrightarrow(\mathbb{C}P^n,h), \] where $g_\epsilon$ is a Berger metric and $h$ is the Fubini–Study metric. For $n\geq3$, we determine all negative and zero eigenvalues and obtain explicit formulas for the Morse index and nullity for every $\epsilon>0$. In particular, for the unit round metric, $\mathrm{Ind}(\pi)=2n+2$ and $\mathrm{Null}(\pi)=n(3n+5)$. We then extend Rivière's rigidity argument from $\mathbb{S}^3\to\mathbb{S}^2$ to higher dimensions. If $\phi:\mathbb{S}^{2n+1}\to\mathbb{C}P^n$ is harmonic and $\mathrm{Ind}(\phi)=2n+2$, then $d\phi_x\circ d\phi_x^\ast$ commutes with the complex structure of $\mathbb{C}P^n$ at every point. Consequently, up to an isometry of the domain, $\phi$ factors through the Hopf fibration as $\phi=P\circ\pi\circ f$, where $P:\mathbb{C}P^n\to\mathbb{C}P^n$ is holomorphic.

Let $x:M^n\to\mathbb{S}^{n+q}$ be a minimal immersion of a complete connected manifold with flat normal bundle, where $n\ge3$ and $q\ge2$. We prove that if the squared norm of the second fundamental form $S$ satisfies $S<2n$, then $M^n$ lies in $\mathbb{S}^{n+1}$. The standard minimal products of three spheres show that the upper bound $2n$ is optimal. We also prove a codimension reduction theorem under the condition $S+\lambda_2\le 2n+\frac{2n}{3n-2}$, where $\lambda_2$ is the second largest eigenvalue of the Gram matrix of the shape operators, implying that $M^n$ lies in $\mathbb{S}^{n+1}$.

We establish Tachibana-type rigidity theorems with the aid of the curvature operator of the second kind (COSK). A sharp estimate that combines the symmetric and commutator actions on Weyl tensors leads to a constant $C_{n}$, with $\frac{C_{n}}{n}\to\frac{5}{4}$, such that a complete Einstein manifold of dimension $n\geq5$ with $C_{n}$-nonnegative COSK has constant nonnegative sectional curvature. We also explore shifted curvature operators of the second kind leading to similar rigidity results, offer an elegant Ricci-corrected Bochner identity for harmonic forms, and classify the complete simply connected locally symmetric Einstein manifolds in the resulting shifted cone. Explicit spaces establish the sharpness of the tensor estimates, and complete lists of the eigenvalues of the COSKs for symmetric spaces identify which of them are borderline cases.

We prove the sharp horizontal isoperimetric inequality in the first Heisenberg group for all sets of finite horizontal perimeter and positive finite volume, with equality precisely for left translates and dilates of the Pansu ball.

It is known that on complete Riemannian manifolds with nonnegative Ricci curvature, the space of harmonic 1-forms of polynomial growth is finite-dimensional. In this paper, we extend this finiteness phenomenon to some manifolds whose Ricci curvature is allowed to be negative. More precisely, under a scale-invariant $L^2$-Sobolev inequality, a weak Hardy inequality, and quadratic Ricci curvature decay with a sufficiently small coefficient, we prove that the space of polynomial-growth harmonic 1-forms is finite-dimensional. We also construct a counterexample showing the necessity of the smallness condition. Our result is closely related to an earlier theorem of L.F.Tam \cite{tam1998note}, where the finite-dimensionality of polynomial-growth harmonic 1-forms was established under a faster curvature decay assumption, without any smallness condition on the coefficient. In comparison, our result treats the critical quadratic decay regime, at the expense of requiring the corresponding coefficient to be sufficiently small. As a byproduct, we obtain an analogous finite-dimensionality result for polynomial-growth solutions to the Schrödinger heat equation, which generalizes the finite-dimensionality results in \cite{lin2019ancient},\cite{colding2021optimal} and \cite{lin2026space} to some extent.

We obtain topological obstructions to the existence of a complete Riemannian metric with uniformly positive scalar curvature on the interiors of certain 4-manifolds using Seiberg–Witten theory. In particular, no punctured blowups of a homotopy K3 manifold admit complete metrics of uniformly positive scalar curvature. Further, let $X$ be a compact oriented smooth manifold of dimension $5$, $6$, or $7$. We show that if $\operatorname{Int}X$ admits a complete metric of uniformly positive scalar curvature, then $X$ admits a positive scalar curvature metric with minimal boundary. In dimensions $6$ and $7$, the metric may moreover be chosen to be a product near the boundary. In dimension 5, under additional hypotheses and after possibly changing the smooth structure on $X$, one can likewise obtain a positive scalar curvature metric that is a product near the boundary.

We establish curvature-based bounds for the $p$-capacity of convex domains and related capacity–area inequalities. Let $K$ denote the Gauss–Kronecker curvature of $\partial\Omega$. In Euclidean space, if $ 0<K_-\leq K\leq K_+, $ then, for every $1<p\leq \frac{n+1}{2}$, $$ \operatorname{Cap}_p\left(B(K_+^{\frac{1}{1-n}})\right) \leq \operatorname{Cap}_p(\Omega) \leq \operatorname{Cap}_p\left(B(K_-^{\frac{1}{1-n}})\right). $$ The lower comparison remains valid for the full range $1<p<n$. Moreover, the exponent range in the upper comparison is optimal in general: when $n\geq 4$ and $\frac{n+1}{2}<p<n$, the upper comparison may fail. Thus $p=\frac{n+1}{2}$ is the critical exponent for the curvature-radius upper comparison. Under the pinching condition $ \frac{K_-}{K_+}=\rho\in (0, 1], $ we further obtain an explicit scale-invariant capacity–area lower bound for every $1<p<n$. In the special case $n=3$ and $p=2$, $$ \frac{\operatorname{Cap}_2(\Omega)} {\sqrt{|\partial\Omega|}} \geq \frac{4\sqrt{\pi}\,\rho^{1/4}} {1+\sqrt{\rho}}, $$ which implies the Pólya–Szegö inequality whenever $$ \rho\geq \rho_*:= \left( \frac{\pi^2-4-\pi\sqrt{\pi^2-8}}{4} \right)^2 \simeq 0.15484. $$ On a Cartan–Hadamard manifold with $\operatorname{Sec}\leq -k^2<0$, we obtain a $p$-capacity lower bound retaining all principal curvatures of the boundary, together with corresponding Gauss–Kronecker curvature estimates and a rigidity characterization of the equality case by a warped-product exterior geometry.

In this paper, we study Ricci flows with uniformly bounded scalar curvature. On such Ricci flows we establish uniform $L^2$ Riemannian curvature bounds and $L^4$ Ricci curvature bounds up to dimension 7. These estimates improves $L^{2-\varepsilon}$ curvature bounds established by Bamler and $L^2$ curvature bounds in dimension 4 established by Bamler and Zhang. More generally, we establish uniform $L^2$ curvature bounds on $n$-manifolds satisfying an integral $L^p$ Ricci curvature bound for $p> n/2$, a uniform volume lower bound, and a strong $\varepsilon$-regularity assumption. This result extends the celebrated $L^2$ curvature estimates of Jiang and Naber, which were established under pointwise two-sided Ricci curvature bounds. To overcome the lack of pointwise control, we develop an $L^p$ adaptation of the neck decomposition theorem and establish a superconvexity estimate for the curvature on neck regions. Furthermore, we demonstrate that our structural assumptions are dynamically natural on Ricci flows with bounded scalar curvature. As an application, by combining our estimate with the a priori $L^{4-\varepsilon}$ curvature radius bound of Ricci flow established by Bamler, we obtain uniform $L^2$ curvature bounds on Ricci flows with bounded scalar curvature of dimension $n \le 7$.

We prove harmonicity for nonshrinking algebraic generalized Ricci solitons and for all algebraic generalized Ricci solitons on solvable Lie algebras. In the solvable setting, we characterize expanding solitons by normal extensions of generalized nilsolitons, and derive a sharp rank bound.

We prove that the set of real $\mathrm{SL}_n$-opers forms a discrete subset of the moduli space of irreducible flat connections.

We introduce contact monopoles on contact metric $5$-manifolds and study their basic geometric properties. We construct natural examples from Boothby–Wang fibrations over Kähler surfaces and relate the equation to contact instantons on Sasakian $5$-manifolds. In particular, we show that anti-self-dual contact instantons with a parallel Higgs field provide contact monopoles. We also prove a compact rigidity result: on compact contact metric $5$-manifolds, every smooth contact monopole has parallel Higgs field and satisfies the reduced curvature equation $F_A\wedge\omega=0$. In the Sasakian case, this relates compact contact monopoles to anti-self-dual contact instantons under the usual transverse type condition. We also explain why the associated deformation complex is not elliptic in the usual sense. We then focus on unit tangent sphere bundles of oriented Riemannian $3$-manifolds, endowed with their canonical contact metric structures. In this setting, we derive a local form of the contact monopole equation with respect to the canonical adapted frame and prove that pure pullback configurations from the base are rigid. Finally, for the homogeneous Sasakian manifold $T_1S^3\cong SO(4)/SO(2)$, we use Wang's theorem to reduce the invariant contact monopole equation to finite-dimensional algebraic systems and classify the invariant solutions for homogeneous $U(1)$- and $SO(3)$-bundles. In the abelian case, the classification yields non-flat homogeneous $U(1)$ contact monopoles for nonzero topological charge. In the non-abelian case, all weights except $m=1$ reduce to the maximal torus, while the weight $m=1$ gives a genuinely non-abelian branch of homogeneous $SO(3)$ contact monopoles.

We show that quasi-Poisson and quasi-BV structures on (super)manifolds $M$ admitting an action of a Lie (super)group $G$ can be identified with a special class of homotopy Poisson and respectively $BV_\infty$ structures on the associated differential (super)manifolds of the form ${\mathcal M}:=M\times \Pi g$, $g$ being the Lie (super)algebra of $G$. The construction is based on the proven existence of an essentially unique family of higher Jacobiators for any Lie (super)algebra $g$ with an invariant (odd or even) scalar product. We apply these arguments to the moduli spaces $M$ of flat $G$-connections on a surface with boundaries using Fock-Rosly construction.

We prove that, in most moduli spaces of orientable hyperbolic surfaces of finite area, there exist surfaces all of whose covers of degree two have a new small eigenvalue. On the other hand, we show that four punctured spheres always admit a double cover without new small eigenvalues and that there exist closed surfaces of any genus with the same property. We prove the last two of these results using a version of Cheeger's inequality adapted to double covers.

This paper is primarily concerned with the existence of a symplectic $4$-manifold, homeomorphic but not diffeomorphic to $CP^2$, which admits a nontrivial finite order symplectomorphism. We reduced the existence of such an exotic $CP^2$ to a question in contact geometry. More concretely, we showed that there are precisely $12$ plumbed manifolds, each equipped with a canonical contact structure associated to the plumbing, such that an exotic $CP^2$ with a nontrivial finite order symplectomorphism exists if and only if one of the $12$ contact plumbed manifolds admits a certain specific symplectic filling. As the main technical result, we completely determined the order and local representations of a pseudofree symplectic $Z_p$-action for an odd prime $p$ on a rational homology $CP^2$ with positive canonical class. Our result suggests that potentially, there are symplectic $Z_p$-actions which are not a finite automorphism of a fake projective plane. Moreover, we introduced a construction, which, under the minimal assumptions on the symplectic fillings of the $12$ contact plumbed manifolds, produces a symplectic $4$-manifold on the Bogomolov-Miyaoka-Yau line via a reverse-engineering process.

We establish an a priori $C^2$-estimate for complex Hessian equations on compact Hermitian manifolds. As an application, we obtain a regularity result for solutions to complex Hessian equation with densities of the right hand side depending on the solution. This generalizes Székelyhidi–Tosatti's result to the setting of complex Hessian equations on Hermitian manifolds.

We extract rank and regularized exponents from cyclic generic factors of the quantum connection and show that these data persist under the birational comparisons considered here. Applications include showing non-rationality of cubic threefolds, ordinary Gushel–Mukai threefolds, very general quartic fourfolds, and products of the threefolds with $\PP^1$.

Let $(M^n,g_0)$ be a closed connected Einstein manifold with positive scalar curvature, where $n\ge6$. Using an Obata-type identity, we prove that every smooth metric conformal to $g_0$ with constant sixth-order $Q$-curvature is Einstein.

We study a general class of fully non-linear elliptic Hessian equations on compact Riemannian manifolds. In particular, we replace the standard assumption that the equation is a symmetric function of the eigenvalues of the Hessian made by Caffarelli, Nirenberg and Spruck by a weaker one. After deriving a priori estimates for solutions under suitable assumptions on the equation, we solve it with a continuity method.

We study regularity of bounded weak solutions to complex Monge-Ampère equations on compact Hermitian manifolds with right-hand side possibly decreasing in the unknown. Our proof relies on the domination principle and a priori estimates, partially motivated by the Monge-Ampère eigenvalue problem. Our main result roughly says that any bounded solution is smooth in the regular locus of the data. When the right-hand side is strictly positive and smooth, we recover known results by Nie, Ko lodziej-Nguyen for the Hermitian case and Székelyhidi-Tosatti for the Kähler case, which rely on the regularizing property of the Kähler-Ricci flow.

Let $(M^n,g_0)$ be a closed hyperbolic manifold, $n\geq3$. We compute the Hessian of volume entropy at $g_0$ under the constraint $Sc_g=-n(n-1)$ and express it as a quadratic form on transverse-traceless symmetric two-tensors. In dimension three, the Hessian is negative definite, and every sufficiently $C^\infty$-close metric with $Sc_g\geq-6$ has entropy at most $2$, with equality precisely for pullbacks of $g_0$. In dimensions $n\geq4$, a universal spectral threshold determines the Hessian sign. Nonzero trace-free Codazzi tensors give positive directions, and hyperbolic bending provides smooth constrained entropy saddles in every such dimension.

Sasakian manifolds, a subclass of contact metric manifolds, play a central role in modern differential geometry and mathematical physics, where they arise naturally in geometric mechanics, CR geometry, and supersymmetric field theories. The weak contact metric (w.c.m.) structure generalizes the contact metric structure and provides a broader framework for studying contact geometry and its applications. A fundamental problem is to understand how curvature conditions constrain the underlying contact metric structure and, in particular, how they distinguish Sasakian geometry from its weaker variants. Using the partial Ricci flow, we characterize Sasakian structure (among w.c.m. manifolds) under the $(\kappa,\mu)$-nullity condition related to curvature. For $\kappa<1$ we find conditions under which a w.c.m. manifold admits a bi-Legendrian structure, and for $\kappa=\mu=0$ establish splitting and classification results.

It was recently proved that an embedded minimal annulus with free boundary in the unit ball $\Bbb B^3$ of the Euclidean space is, up to a rotation, the critical catenoid. We prove that an immersed, possibly branched, free boundary minimal annulus in $\Bbb B^3$ with embedded boundary components is necessarily free of branch points and is embedded. This shows the uniqueness of the critical catenoid holds under these weaker hypotheses. We then apply these results to prove that if $\Omega$ is an annulus in $\Bbb S^2$ for which there exists a smooth function satisfying the overdetermined eigenvalue problem \begin{equation*} \begin{cases} \Delta u+ 2u = 0 \quad \text{on} \quad \Omega\\ \qquad \quad u=0 \quad \text{in}\quad\partial\Omega\\ \quad\,\,\,\, |\nabla u|=1 \quad \text{in}\quad \partial\Omega. \end{cases} \end{equation*} then $\Omega$ is, up to rotation, a rotational annulus with equatorial symmetry.

A pseudo-Riemannian manifold is called geodesic orbit (GO) if every geodesic is an orbit of a one-parameter group of isometries. For Riemannian $H$-type groups, the GO property was completely characterized by A. Kaplan and C. Riehm in terms of the dimension of the centre and the structure of the underlying Clifford module. We solve the corresponding problem for pseudo $H$-type groups $N_{r,s}$, that is, 2-step nilpotent Lie groups whose Lie algebra $\mathfrak n_{r,s}=\mathfrak z\oplus\mathfrak v$ is built from an admissible module $\mathfrak v$ of the Clifford algebra $\mathrm{Cl}(\mathbb R^{r,s})$, endowed with the left-invariant metric whose restriction to the centre is indefinite ($s\ge 1$). We prove that $N_{r,s}$ is naturally reductive if and only if $(r,s)\in\{(0,1),(1,2)\}$, and then it is GO for every admissible module. We also prove that $N_{3,4}$ is GO, but not naturally reductive, exactly when $\mathfrak v$ is a minimal admissible module, and that in all remaining cases $N_{r,s}$ is not GO. Thus, in contrast with the Riemannian situation, where a seven-dimensional centre admits GO metrics on isotypic modules of dimensions 8, 16 and 24, the exceptional pseudo-Riemannian case is GO only for the smallest possible module. For $N_{3,4}(\mathfrak v_{\min})$ we explicitly construct the one-parameter isometry groups generating all geodesics, including the null ones, and describe the affine family of their generators.

We prove the LeBrun–Salamon conjecture for positive quaternionic Kähler manifolds of quaternionic dimension n <= 14. To do so, we first show that the isometry group Isom(M) of any minimal counterexample M has rank at most one, hence dimension at most three. On the other hand, we consider a characteristic number which equals dim Isom(M) - (n+3) if n is even and dim Isom(M) - (n+1) if n is odd, and show that this index is a sum of positive quantities when n <= 14, yielding a contradiction. The toolset used to show positivity enlarges in stages. Projection curvature moments suffice through n=8; positive-semidefinite curvature moments extend the argument through n=10; and signed scalar orbital moments give the new inequalities needed for n=11,12. Scalar orbital moments alone fail at n=13; combining them with Hodge–Riemann positivity for squares of cohomology classes extends the proof to n=13,14. We give exact separating obstructions explaining the changes of method.

We show that a convex surface in Euclidean 3-space is a sphere if the area cut off from it by a slab of some fixed width depends only on the direction of the slab. This result generalizes theorems of Blaschke and Stamm, which assumed the slab-area property for every width. It holds also for any smooth closed embedded surface. The proof is based on an averaging identity, which relates slab areas to the Newtonian potential of the surface, the electrostatic characterization of spheres by Mendez and Reichel, and analytic microlocal properties of the Radon transform, which are used to show that the slab-area property forces real analyticity, via theorems of Kashiwara and Boman.

We study four of Manton's five vortex equations on a Riemann surface $X$ in relation to anti-self-dual connections. From a solution of each vortex equation, we construct an equivariant holomorphic bundle over $X$$\times$$\mathbb{P}^1$, $X$$\times$$\Delta$, or $X$$\times$$\mathbb{C}$, together with an invariant Hermitian or pseudo-Hermitian metric whose associated connection is anti-self-dual. The equivariant structures are given by the actions of $SU(2)$, $SU(1,1)$, and $SE(2)$, respectively. Conversely, we show that an equivariant holomorphic bundle equipped with an invariant (pseudo-)Hermitian metric gives rise to a solution of the corresponding vortex equation when the associated connection is anti-self-dual.

Let $(M^n,g)$ be a complete connected noncompact smooth Riemannian manifold without boundary, with $n\ge3$ and positive scalar curvature. We prove that $M$ admits a complete smooth metric of uniformly positive scalar curvature if $$ \liminf_{d_g(o,x)\to\infty}d_g(o,x)^2R_g(x)>\frac{n-1}{n} $$ for a fixed point $o\in M$. This extends earlier existence results for orientable manifolds, due to Chen in dimension three and to the authors in dimensions $4\le n\le7$, to every dimension $n\ge3$ without an orientability assumption.

We extend the Wentzell operator to differential forms on compact manifolds with boundary, investigate its spectrum, and establish a variational characterization of the first eigenvalue. We explicitly compute the eigenvalues on the unit ball in Euclidean space. We also derive several eigenvalue estimates for the first eigenvalue, some of which are shown to be sharp. Finally, we obtain a lower bound for the gap between the lowest eigenvalues in consecutive degrees.

We show that the fundamental group is not locally stable under non-collapsed Gromov–Hausdorff convergence, even among closed Ricci-flat Kähler four-manifolds. More precisely, we construct families of Ricci-flat Kähler metrics on $K3$ surfaces and Enriques surfaces that converge to the same compact flat orbifold. Using the same mechanism, we further construct two families of closed locally hyperkähler Ricci-flat four-manifolds, with fundamental groups $\mathbb Z_2$ and $\mathbb Z_2\times\mathbb Z_2$, respectively, which converge to the same compact flat orbifold.

Constant curvature is one of the most fundamental intrinsic conditions in submanifold geometry. Investigating the existence and classification of biharmonic submanifolds under intrinsic curvature constraints is generally difficult, since the biharmonic equation is expressed in terms of extrinsic data. In this paper, we obtain a complete classification of biharmonic Lagrangian surfaces with constant Gaussian curvature in 2-dimensional complex space forms. As a consequence, we classify the biharmonic Lagrangian surfaces with flat normal bundle and we prove that pseudo-umbilical biharmonic Lagrangian surfaces are necessarily minimal.