Papers

We review the definitions of the Rumin complex and analytic contact torsion for contact manifolds. We then compute the contact torsion for symmetric contact manifolds. To this end, we use a representation-theoretic description of the relevant operators and spaces of differential forms. Our computation of the contact torsion for symmetric contact manifolds relies on a recent result by Rumin that reduces the problem to harmonic forms. Following a construction by Boothby and Wang, the contact manifolds under consideration are total spaces of $S^1$-principal bundles over Kähler manifolds. This allows us to interpret the harmonic forms as Dolbeault cohomology with coefficients in a certain holomorphic line bundle of the base Kähler manifold, thereby simplifying the computation by allowing us to use results from complex geometry. Furthermore, we generalise Rumin's result to the equivariant case and, using the previous considerations, determine the equivariant contact torsion for the case of isolated fixed points.

This is the second article of a sequence of research on the local rigidity of constant mean curvature (CMC) hypersurfaces in space forms. In the previous one, we studied the local rigidity of CMC hypersurfaces whose the number of the distinct principal curvatures satisfies $g\leq 3$. In this paper, we study the local rigidity of CMC hypersurfaces with $g\geq 4$. When $g>4$, we prove that if the $k$-order mean curvatures $H_k$, $(k=2,\cdots, g-1)$ are constant and there exist enough multiple principal curvatures, then the CMC hypersurface is an isoparametric hypersurface. When $g=4$, if $H_2$ and $H_3$ are constant, then the CMC hypersurface is an isoparametric hypersurface.

In this note, we study volume comparison for 3-manifolds with 2-Ricci curvature lower bound. We establish monotonicity and rigidity for geodesic sphere area and ball volume, and derive the pointed Gromov-Hausdorff precompactness under a uniform lower bound on the conjugate radius for 3-manifolds with 2-Ricci curvature lower bound. We also discuss the almost preservation of 2-Ricci curvature lower bound and the local non-collapsing propagation along Ricci flow. Finally, we point out a topological rigidity result on nonnegative scalar curvature.

We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

We prove a Jacobian estimate underlying both the Heintze–Karcher comparison and the Alexandrov–Bakelman–Pucci method for submanifolds of arbitrary codimension. The estimate retains the contributions of ambient curvature along geodesics emanating from the submanifold and requires no curvature sign assumption. As applications, we obtain a quantitative Fenchel–Willmore inequality with explicit curvature remainders under nonnegative $n$-intermediate Ricci curvature, Michael–Simon Sobolev and isoperimetric inequalities under nonnegative $(n-1)$-intermediate Ricci curvature, and corresponding extensions under quadratic curvature decay. In each result, the required intermediate Ricci curvature condition depends only on the dimension of the submanifold and is independent of its codimension.

In this paper, we study the boundary regularity of harmonic map heat flows from a bounded domain in a metric measure space with generalized Ricci curvature bounded from below into a non-positively curved metric space.

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMCP^+(K,N)$. The cost for this problem is the time separation function $\ell(x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling from $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we show that the Lorentzian timelike measure contraction property $TMCP^+(K,N)$ is equivalent to the $TMCP^+_{rLip}(K,N)$ condition, where the curvature dimension conditions are defined on gradient flow curves of reverse 1-Lipschitz functions.

Control allocation for multirotors with bidirectional propellers is commonly formulated in signed-thrust variables, where the wrench map is linear. The signed-quadratic map from physical rotor speed to thrust, however, is not a local diffeomorphism at zero speed. This work derives two distinct consequences. Under nonredundant full actuation, a single-propeller reversal removes one instantaneous task direction; hence, no global continuously differentiable exact static allocator exists over the complete task space. Under redundant actuation, the physical task map may remain regular, yet a transverse pseudoinverse zero crossing requires an unbounded rotor-speed derivative. We define differential realizability as regularity of the physical lift of an actuator-output section, derive exact and first-order validity conditions, and distinguish structural rank loss from an allocator- induced rate singularity. A local nullspace deformation repairs isolated pseudoinverse reversals, while a global fixed-orthant construction establishes existence of regular sections at the cost of persistent task-preserving internal actuation.

Let $2\leq k<n/2$ and let $(M^n,[g])$ be a smooth, closed, connected, and locally conformally flat Riemannian manifold with a $k$-admissible metric in the conformal class $[g]$. We prove a stability result of the $\sigma_k$-curvature inequality on $M$, in the sense that if equality is almost satisfied for some conformal metric, then its conformal factor is close to a minimizer of the inequality. Closeness is measured quantitatively in terms of Sobolev norms of the conformal factor, namely with respect to the $W^{1,2}$- and the $W^{1,2k}$-norm. In the non-degenerate case, these norms come with optimal exponents $2$ and $2k$, respectively, whereas in general the exponents are $2+\gamma$ and $\max\{2k,2+\gamma\}$ for some $\gamma\geq 0$ originating from a Łojasiewicz inequality. This extends a previous result by Frank and the author from $k=2$ and the sphere to $2\leq k<n/2$ and the full class of manifolds originally considered by Viaclovsky. It also extends a previous result by Engelstein–Neumayer–Spolaor from $k=1$ to the setting of fully non-linear scalar curvatures.

In this paper, we give a negative answer to Wang's conjecture. For every $n\geq 3$, there exist $\varepsilon>0$ and $\delta \in (0,1)$ such that, for all $q$ and $\lambda$ satisfying \[ \frac{n}{n-2}-\varepsilon<q\le \frac{n}{n-2}, \qquad \frac{1}{q-1}\cdot \delta<\lambda\le \frac{1}{q-1}, \] there exists a bounded Euclidean domain \(\Omega\subset\mathbb{R}^n\) with principal curvatures bigger than \(1\) for which the following nonlinear Robin problem admits a nonconstant positive solution: \begin{align*} \begin{cases} \Delta u=0, & \text{in}\ \Omega,\\[2mm] \dfrac{\partial u}{\partial\nu}+\lambda u=u^q, & \text{on}\ \partial\Omega. \end{cases} \end{align*}

We establish one form of the vortex filament conjecture for the three-dimensional Gross-Pitaevskii equation. Given any smooth closed embedded binormal flow of curves on a compact time interval, we construct, in the small-core limit $\varepsilon\to0$, a family of exact solutions whose degree-one vortex filaments converge uniformly to the prescribed flow. Near the filament the solutions have the standard planar vortex profile, while away from it their phase gradients converge to the associated Biot-Savart field. We also derive a refined modulation law for the vortex filament.

Z/2 harmonic 1-forms arise naturally as singular limits in gauge theory and calibrated geometry, but the topology that can occur in their singular sets is poorly understood. In this paper we study the flexibility of these singular sets when the ambient Riemannian metric is allowed to vary. We prove that every compact smoothly embedded codimension-two submanifold with trivial normal bundle in a Euclidean space of dimension at least three can be realized as the singular set of a nondegenerate Z/2 harmonic 1-form for a complete metric that is Euclidean outside a compact set. As an application, we use Calabi surgery to construct Z/2-harmonic 1-forms with prescribed local singular sets on closed manifolds of positive first Betti number.

We prove that any embedded genus zero free boundary minimal surface in the ball has first Steklov eigenvalue $1$. This implies the uniqueness of the critical catenoid.

We study the classical problem of sectional curvature pinching in the class of 2-step nilmanifolds, which are necessarily of mixed curvature. We show that the pinching constant of any 2-step nilmanifold lies in the compact interval $[-3, -\frac{3}{2}]$. The upper bound $-\frac{3}{2}$ is achieved by the complex Heisenberg group $\mathrm{Heis}_3(\mathbb{C})$ with a Ricci soliton metric. The upper bound exhibits rigidity: if a simply connected 2-step nilmanifold $N$ has the pinching constant $-\frac{3}{2}$, then $N$ admits a Ricci soliton complex Heisenberg group as a totally geodesic subgroup. On the other hand, the lower bound satisfies non-rigidity: any 2-step nilpotent Lie group admits a metric with pinching constant $-3$. This is derived by showing that there is an open neighborhood $U$ of $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$ in the space of $n$-dimensional 2-step nilmanifolds such that the pinching constant is $-3$ on $U$, and any 2-step nilpotent Lie group $N$ has a metric $g$ such that $(N,g)$ lies in $U$. In fact, if $N$ is not isomorphic to $\mathrm{Heis}_3(\mathbb{R})\times \mathbb{R}^{n-3}$, then there is a curve $g_t$ of metrics on $N$ with $(N,g_t)\in U$, showing that there are uncountably many left-invariant metrics on $N$ such that the pinching constant is $-3$. An algebraic characterization of a 2-step nilpotent Lie group that admits a metric with the pinching constant $-\frac{3}{2}$ is also given, and the pinching constants of various examples are computed.

In this article, we investigate rigidity results for constantly curved minimal immersions of the two-sphere $S^2$ into the unitary group $\mathrm{U}(n)$. Using loop group methods for harmonic maps, we establish a correspondence between such immersions and a distinguished class of constantly curved holomorphic immersions of $S^2$ into finite-dimensional Grassmannians. In the case $\mathrm{U}(3)$, we classify the constantly curved minimal immersions of $S^2$ with uniton number one and prove that, under a natural unramifiedness condition, those of uniton number two are $S^1$-invariant; as a consequence, every constantly curved totally unramified minimal immersion $S^2\to \mathrm{U}(3)$ of uniton number two is unitarily congruent to the composition of the first Gauss map of the Veronese curve in $\mathbb{C}P^2$ with the Cartan embedding $\mathbb{C}P^2\hookrightarrow \mathrm{U}(3)$.

We establish a priori interior $C^{1,\alpha}$ and $C^{2,\alpha}$ Schauder estimates for a class of $Q$-valued quasilinear elliptic systems which are perturbations of the Laplace system, for arbitrary multiplicity $Q$, domain dimension $n\geq2$, and target dimension $k$. The $C^{1,\alpha}$ estimate generalises the work of Simon and Wickramasekera in \cite{SW16} on $2$-valued solutions of linear systems and concerns weak solutions of divergence-form systems, while the $C^{2,\alpha}$ estimate concerns strong solutions. In dimension $n=2$, both estimates hold for every $0<\alpha<1/Q$. In dimensions $n\geq3$, there exists $\delta=\delta(n,k,Q)>0$ such that both estimates hold for every $0<\alpha<\delta$. As applications, we obtain a small-slope Schauder estimate and a small-slope Bernstein theorem for $Q$-valued maps whose graph varifolds are stationary. Combining the Schauder estimate with the recent branch-set stratification theory of Krummel–Minter–Wickramasekera in \cite{KMW26}, we further prove that, for every $\gamma>0$, the branch set $\mathcal B_u$ of any $C^{1,\gamma}$ $Q$-valued map $u$ whose graph varifold is stationary satisfies $\dim_{\mathcal H}\mathcal B_u\leq n-2$. This bound is sharp already in codimension one.

We construct an explicit one-parameter family of smooth metrics on the Gromoll-Meyer exotic seven-sphere, converging in $C^\infty$ to a fixed further Cheeger deformation of the Eschenburg-Kerin metric and having strictly positive sectional curvature for all sufficiently small positive parameter values. The first perturbation preserves the totally geodesic flats of one zero-plane family while making curvature positive near the other; the second removes the remaining zero curvature. The metric and the proof were discovered by the Odin Automatic AI Research Agent.

We formulate the linearization of the fourth Painlevé equation as a linearized Hamiltonian system and transform the scalar variational equation to normal form, with the first-derivative term eliminated. From the canonical fundamental solutions of the Jimbo–Miwa Lax pair, we construct quadratic expressions $Q_{IV}$ and $R_{IV}$ that satisfy the identity $\partial_x^2Q_{IV}-U_{IV}Q_{IV}=\partial_\lambda R_{IV}$. Here $U_{IV}$ is the coefficient in the normal-form equation. A rapid-decay cycle with a vanishing boundary term yields an integral solution of the linearized equation. Every nontrivial such cycle has asymptotic branches in at least two distinct Stokes sectors; otherwise the integral vanishes by Cauchy's theorem. A second rapid-decay homology class gives another integral solution. The Wronskian of the two integral solutions depends holomorphically on the monodromy data, so a single nonzero value implies generic linear independence. For one set of complex data, direct quadrature gives numerical evidence of a nonzero Wronskian. We also derive integral formulas for variations of the Stokes multipliers, the connection matrix, and the local monodromy, and verify the contour representation by substitution into the linearized equation.

We relate the spectrum of singularities to the geometry of degenerating Calabi–Yau manifolds through holomorphic analytic torsion. We express the leading asymptotic coefficients of the analytic torsion of holomorphic differential forms in terms of Hodge-theoretic invariants of vanishing cycles. For isolated hypersurface singularities, these formulas lead to the formulation of new higher Durfee–Saito conjectures bounding Hodge-filtration dimensions by the Milnor number and Eulerian numbers. We also prove that, for isolated quasi-homogeneous singularities, the spectral measure is dominated in convex order by the corresponding Irwin–Hall distribution. Applying the torsion formulas to the BCOV invariant yields a computable obstruction to birational smooth fillings. For isolated singularities, its local contribution is expressed in terms of Hertling's spectral variance and a Bernoulli sum determined by monodromy. As applications, we rule out birational smooth fillings for projective Calabi–Yau degenerations with smooth total space and singular central fiber having only ADE or terminal Brieskorn–Pham singularities, in relative dimension at least three, extending results of Voisin.

We use the Labourie–Wentworth's formula and the thermodynamic formalism to show that, along the slice constructed by Bronstein–Davalo, the logarithmic top-eigenvalue length spectrum has a uniformly positive second variation near the Barbot representation. As a major application, we show that every closed surface group admits a non-Hitchin Borel Anosov representation into $\mathrm{SL}_{3k}\mathbb{R}$ for every $k\geqslant 1$. In particular, we obtain the first such examples in the even dimensions $6k$. We also study the local behavior of related objects of this slice around the Barbot representation, including the Lyapunov exponent of the flat bundle, the Hausdorff dimension of the limit set, and the Hilbert entropy of the representation.

Let $X:M^n\to\R^{n+1}$ be an $n$-dimensional complete self-shrinker. We obtain a complete classification of complete self-shrinkers with positive constant scalar curvature. More precisely, we prove that the round sphere $S^n(\sqrt n)$ and the standard generalized cylinder $S^k(\sqrt k)\times \R^{n-k}$ for $2\leq k\leq n-1$ are the only complete self-shrinkers with positive constant scalar curvature. The key difficulty is to characterize the residual case in Cheng-Li-Wei \cite{CLW}: $R>0$, $S<1$, and $\sup_M S=1$, where $R$ and $S$ denote the scalar curvature and the squared norm of the second fundamental form, respectively. For this case, it seems a hard task that the generalized maximum principle yields a useful information. In order to overcome this substantial difficulty, our key ingredient is to get a uniform positive lower bound for the Bakry-Émery Ricci curvature so that we can make use of the comparison theorem of Wei-Wylie \cite{WeiWylie} to conclude that the Gaussian volume is finite. Furthermore, the gap theorems on $S$ are given.

In this paper, we continue our recent study on anisotropic minimal surface equation with free boundary condition in Wang-Wei-Xia-Zhang (Arch Ration Mech Anal 250:42, 2026). The first main result solves the corresponding mixed boundary value problem. As an application, we prove the following Bernstein-type theorem: any anisotropic minimal graph over $\mathbb{R}^3_+$ with free boundary must be flat. These extend the classical results of Simon (Indiana Univ Math J 25:821–855, 1976; Math Z 154:265–273, 1977) to an appropriate free boundary setting.

In this work, a sharp Kähler spectral almost-rigidity theorem was established, resolving Conjecture 1.8 of Chu–Wang–Zhang. For compact Kähler manifolds satisfying $\Ric(\omega)\geq\omega$, pinching the first $n^2+3$ nonzero complex eigenvalues to one forces the manifolds to be Gromov Hausdorff close to normalized complex projective space and determines the biholomorphism type. The smaller index $n^2+1$ is the sharp threshold for noncollapsing. More generally, if a normalized measured limit has essential real dimension $r$, then the multiplicity of the critical eigenvalue is at most $r+\lfloor r/2\rfloor^2$, with equality attained in every dimension. A uniform energy estimate for kernel projections of complex gradient Gram matrices leads to a Lie algebra action on the whole spectral resolution.

In this paper, we study the ergodicity of geodesic flows on closed rank-one manifolds of nonpositive sectional curvature. We introduce the infinite-order vanishing set of the Gaussian curvature in dimension two and of the fiberwise second moment of the reduced Jacobi determinant in higher dimensions. We bound the Liouville measure of the singular set in terms of the volume of this subset and, for surfaces, obtain a corresponding Hausdorff dimension bound. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic. We also give a geometric criterion for finitely many exceptional regions whose fundamental groups have virtually Abelian images of rank smaller than the dimension of the manifold. Under strict negativity of sectional curvature outside these regions, we obtain a Hausdorff dimension bound for the singular set and prove that the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real-analytic metrics, we characterize the singular set as the zero set of a nontrivial real-analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without further assumptions.

Yau asked whether every complete Riemannian manifold with bounded Ricci curvature and positive injectivity radius admits an isometric embedding into some Euclidean space with bounded mean curvature. We construct a counterexample: a smooth complete metric on $\mathbb{R}^4$ with bounded Ricci curvature, positive injectivity radius, and unbounded full curvature. The Gauss equation then forces every $C^2$ isometric immersion of this metric into a finite-dimensional Euclidean space to have unbounded mean curvature. The metric is assembled from disjoint curvature peaks whose Ricci contractions stay uniformly bounded. A short-geodesic curvature estimate and the Jacobi equation give the uniform injectivity-radius bound. The construction works in every dimension at least four, and in dimension four the Ricci curvature may be chosen to tend to zero at infinity.

Let $M^n$ be a closed minimal hypersurface of the unit sphere with constant squared norm $S$ of the second fundamental form. For $n\geq4$, we prove that $S>n$ implies $S>85n/59$, improving the bound $10n/7$ of Suh and Yang. No constancy assumption is imposed on the cubic trace $f_3$. The proof combines a symmetrized Hessian estimate, an integral identity for a mixed moment, and a weighted estimate for the repeated-index components of the covariant derivative of the second fundamental form. An exact polynomial argument treats the entire excluded interval. We also prove an explicit positive lower bound for the average of the normalized quadratic defect $P/S^2$, where $P=f_4-f_3^2/S-S^2/n$. This second estimate holds in every dimension $n\geq2$ and gives a quantitative obstruction to concentration near the locus of two principal curvatures.

In this paper, we study the harmonicity and existence of algebraic generalized Ricci solitons. Firstly, we characterize harmonic torsion of algebraic generalized Ricci solitons on arbitrary metric Lie algebras by an identity involving the Killing form. In particular, positive semidefiniteness of the Killing form implies harmonicity without a unimodularity assumption. Then, we construct generalized nilsolitons with nonzero torsion on indecomposable three-step nilpotent Lie algebras admitting no classical nilsoliton, in dimension seven and in dimensions $6m+d$ for $m>d\geq1$. Furthermore, we provide a spectral obstruction for nilpotent Lie algebras with abelian derived algebra and an obstruction based on the action of derivations on the quotient by the center. Combining these obstructions with explicit constructions, we prove that the filiform Lie algebra $\mathfrak m_2(n)$, $n\geq5$, admits a generalized nilsoliton if and only if $5\leq n\leq8$.

We survey a geometric approach to the Navier-Stokes equations developed by the author over the past fifteen years. The central object is a Riemannian space of fourteen dimensions whose metric is Ricci-flat precisely on the solutions of the Navier-Stokes system, so that the behaviour of a viscous incompressible fluid becomes a property of curvature. The space decomposes into a flat six-dimensional part and two dual quadruples of coordinates, Eulerian and Lagrangian, which exhibits the two classical descriptions of a fluid as two halves of a single geometry. We also recall the associated six-dimensional metric, whose integrability condition is the incompressibility of the fluid; the link with the projective geometry of second-order ordinary differential equations in the sense of E. Cartan; and the use of Cartan's invariants to construct a three-dimensional analogue of the Taylor-Green vortex. An application of the same construction to the equations of rotation of a rigid body is indicated.

In this article, we study Ricci-Yamabe solitons on the Lie group $\mathrm{Sol} \times \mathbb{R}^n$ equipped with a natural left-invariant Riemannian metric, explicitly determining the vector fields that characterize them. We then deduce that, in the case of a Ricci soliton, it is expanding, whereas in the case of a Yamabe soliton, it is shrinking. Finally, we show that if this Lie group is a gradient Ricci-Yamabe soliton, the vector field belongs to $\operatorname{Span}\{\partial_{t_1}, \dots, \partial_{t_n}\}$, and we explicitly provide the Perelman potential.

Let $X$ be a hyperbolic Riemann surface and let $\mu$ and $\nu$ be laminations on $X$ homotopic to measured foliations with finite Dirichlet integral. We prove that $\mu$ and $\nu$ are filling if and only if there exists a homeomorphism to another Riemann surface $f:X\to Y$ and an integrable holomorphic quadratic differential $q$ on $Y$, unique up to the natural equivalence, such that the push-forward laminations are homotopic to the horizontal and vertical foliations of $q$ respectively. This extends a classical theorem of Gardiner-Masur from closed surfaces to arbitrary surfaces. As well, the dual $\mathbb{R}$-tree interpretation yields the solution of an asymptotic Plateau problem for minimal surfaces in a product of two $\mathbb{R}$-trees. We construct examples such that $f:X\to Y$ is not homotopic to a quasiconformal map, and we present sufficient conditions that ensure it is. We deduce applications to main inequalities for locally quasiconformal maps, harmonic maps between surfaces, and big mapping class groups.