Papers

Let $(X_i,d_i,\mathfrak m_i)$ be compact $\mathrm{RCD}(K,N)$ spaces converging to a space $X$ of rectifiable dimension $m$. Their first Betti numbers can drop by at most $N-m$. When equality holds, the maximal abelian covers have a non-collapsed limit $Y$ carrying a free isometric $\mathbb{R}^{N-m}$-action. We turn this limiting symmetry into fibrations of the approximating spaces. More precisely, after passing to a subsequence, there is an open full-measure set $G\subset X$, containing every regular point, on which $X$ is a topological orbifold and the $X_i$ admit local Seifert fibrations with $(N-m)$-torus fibres. Each finite local group acts on the fibre by translations. If the $X_i$ have no boundary, then $X\setminus G$ has Hausdorff codimension at least two. If $Y$ has no bubbling and $X$ is a smooth closed Riemannian orbifold, the local fibrations may be chosen as restrictions of a single global Seifert fibration. An affine replacement on the smooth manifold cover shows, in addition, that a finite cover of $X_i$ is homeomorphic to a product with $\mathbb{T}^{N-m}$; here we use the classification of affine torus bundles at the Betti number equality of Peng, Wang and Wang. When the base is a closed Riemannian manifold, the maps are torus bundles, confirming a conjecture of Zamora and Zhu for possibly singular RCD total spaces. The new ingredients are a pointwise linearization of the collapsing action at regular orbits, an invariant harmonic transverse coordinate compatible with finite isotropy, and an exactly equivariant orbit coordinate built from an elementary rounding-and-doubling argument in the collapsing deck group.

We study the behavior of isolated singularities of stationary harmonic maps with generic boundary data. For round sphere targets, we prove that, for $4\leqslant n\leqslant7$, every stable stationary harmonic map from a bounded smooth $n$-dimensional domain to round $(n-1)$-sphere with generic smooth boundary data has only radial projections composed with orthogonal transformations as tangent maps at its singularities; for 7-dimensional domains and round $k$-sphere targets with $k\geqslant7$, every stable stationary harmonic map with generic smooth boundary data is smooth. These results follow from a general minimum-index principle for closed real-analytic target manifolds. Under suitable target hypotheses ensuring strong compactness and excluding lower-dimensional singularity models, we show that, in the first domain dimension in which singularities can occur, generic boundary data force the link of every singular tangent map to attain the smallest possible Morse index. This principle applies both to stable stationary harmonic maps and, under the corresponding stronger target hypotheses, to all stationary harmonic maps. If no non-constant tangent map in the relevant class has a link attaining this minimum, generic smoothness follows. As an application, we establish a rigidity at infinity theorem. For $3\leqslant n\leqslant7$, we show that any energy-minimizing map from $n$-Euclidean space to round $(n-1)$-sphere admitting the radial projection as a blow-down limit must itself be a translate of the radial projection.

By the paper \cite{Pet2009upper}, \cite{LiNan2026}, we know that there exists $C(n)>0$, for any complete compact or non-compact Riemannian n manifold $M$ with non-negative sectional curvature, without boundary and any $R>0$, \begin{equation} R^{2-n}\int_{B(p,R)} Scal \le C(n). \end{equation} Based on this result, we will prove that (1) There exists constant $C(n)$. If $M$ is a complete, n dimensional, non-compact Riemannian manifold with non-negative sectional curvature, then for any $p\in M$, $R>0$, \begin{equation} R^{2-n}\int_{B(p,R)} Scal \ dvol \le C(n)(1-v(M)), \end{equation} where $Scal$ is the scalar curvature and $v(M)=\lim\limits_{R\to \infty} \frac{volB(p,R)}{volB(0,R)}$ is the asymptotic volume ratio. (2) There exists constant $C(n)$. If $M$ is a complete, n dimensional Riemannian manifold with sectional curvature $\ge 1$, then \begin{equation} \int_M (Scal-n(n-1)) \ dvol \le C(n)(1-\frac{vol(M)}{vol(S^n(1))}). \end{equation}

In this paper, we prove the equivalence between coerciveness and the existence of a solution to the generalised Monge-Ampère equations for right-Noetherian polynomials satisfying several assumptions. This includes the case of strictly interlacing polynomials. In particular, we have generalised Chu-Lee's result to supercritical but not hypercritical LYZ equations.

In 2012, Iwaniec, Kovalev and Onninen proposed an upper Nitsche–Grötzsch type estimate for harmonic $K$-quasiconformal homeomorphisms between circular annuli: in normalized form, every such map $h:A(1,s)\to A(1,S)$ should satisfy \[ S \leq \frac{K+1}{2}s-\frac{K-1}{2s}. \] The radial and spiral-radial one-mode models explain why this estimate is natural. We show that this one-mode evidence does not extend to the unrestricted non-radial class. For every $1<s<S$, we construct a non-radial harmonic orientation-preserving diffeomorphism $h:A(1,s)\to A(1,S)$ with \[ \norm{\omega_h}_{\infty}<\frac{S-s}{S-s^{-1}}, \] strictly below the conjectural threshold. Thus, the dilatation lower bound predicted by the conjecture fails, even for smooth harmonic diffeomorphisms with prescribed circular boundary components. The construction uses a small high-frequency reparametrization of the outer boundary: it lowers the first Fourier-mode dilatation by order $t^2$, while the compensating high modes are exponentially damped at the inner boundary. We also record structured regimes in which the one-mode estimate survives, including an inner-boundary anti-conformal energy condition, a Fourier leakage criterion, and a low-frequency spectral stability result. In the minimal-surface interpretation, the upper radial model is helicoidal and vertical-periodic rather than single-valued catenoidal; accordingly, the counterexamples give vertical-periodic minimal annuli with slope below the expected helicoidal threshold, and lead to a non-radial extremal problem for harmonic annulus diffeomorphisms.

We establish index estimates for complete two-sided free boundary minimal surfaces in smooth mean-convex domains of $\mathbb{R}^3$ with noncompact boundary. We first prove $3{\rm Ind}_s(\Sigma)\geq 2g+b-1$, where $g$ and $b$ describe the conformal compactification. We then include all interior ends with their multiplicities, without further asymptotic assumptions, and selected boundary ends under an explicit condition ensuring vanishing cutoff errors. The proof combines a localized energy identity with a Riemann–Roch count on the conformal double. A boundary puncture at which the chosen forms are regular eliminates the exceptional space; if poles are allowed at every boundary puncture, this space has dimension at most one. We provide the mixed cutoff construction, examples distinguishing embedded boundary ends from reflected planar ends, and a separate analysis of the conformal Jacobi metric. The latter yields finite Dirichlet energy of the logarithmic conformal factor, but the unrestricted boundary-end estimate remains an open step. The compact-boundary case was treated by Cavalcante, Mendes, and dos Santos.

Nitsche proved that every smooth Jordan curve in $\mathbb{R}^3$ of total curvature at most $4\pi$ bounds a unique minimal disk, which is moreover strictly stable. We prove an analogue of this result for Riemannian $3$-balls with mean convex boundary, under an explicit pinching condition on the negative sectional curvature, together with a bound on the covariant derivative of the Ricci tensor. In this setting, every smooth Jordan curve in the boundary sphere of total curvature at most $4\pi$ bounds a unique embedded minimal disk which is strictly stable.

The present paper studies integral geometry problems on three-dimensional Riemannian balls, where integration is performed over minimal surfaces or, more generally, $\Phi$-surfaces defined by an elliptic curvature functional $\Phi$. The space of all $\Phi$-surfaces spanned by round circles on the boundary is a three-dimensional manifold, which we call the space of circles. We show that, when the metric is $\Phi$-simple - a notion which extends to this setting the notion of simple metrics in the geodesic case -, the Gauss lifts of the $\Phi$-surfaces define a foliation of the unit tangent bundle that should be viewed as a two-dimensional analogue of the standard geodesic foliation. This is achieved by solving a foliated Plateau problem on the ball. We then analyze the associated surface Radon transform corresponding to integration along the surfaces and show that it has a finite-dimensional kernel; we also prove that it is injective for an open and dense set of metrics. In the special case of a foliation by minimal surfaces, we apply these results to solve the following boundary area rigidity problem: does the collection of areas of the minimal surfaces determine the metric up to isometry?

We construct a smooth Riemannian metric with strictly positive sectional curvature on every smooth homotopy seven-sphere. For each smooth structure, we construct positively curved metrics on two seven-dimensional disks whose induced boundary metrics agree under the prescribed attaching map. Under this identification, a gluing theorem then yields the required smooth metric on the closed manifold. The construction applies to all 28 oriented diffeomorphism classes.

Let $\Omega$ be an open bounded connected subset of $\mathbb{R}^N$, $N \geq 2$, of class $C^{2,\alpha}$, for $\alpha \in (0,1)$. Let $p \geq 2$ and $\beta>0$. We prove the symmetry of the solution of the $p$-torsion problem with Robin boundary condition subject to the natural overdetermined condition coming from a shape derivative argument and to an extra condition on $\beta$ and on the minimum of the principal curvatures of $\partial\Omega$. The proof is based on some new integral identities, involving the linearized operator of the $p$-Laplacian applied to the standard $P$-function. In passing we prove some other rigidity results in the spirit of Serrin's and Alexandrov's Theorems.

We prove that every complete nonsteady gradient Kähler-Ricci soliton with constant scalar curvature is rigid. This establishes Cao's rigidity conjecture for Kähler-Ricci solitons in arbitrary dimension and gives the corresponding result for expanding solitons, without additional curvature assumptions. The proof uses a rigidity criterion for gradient Ricci solitons with constant scalar curvature, expressed by the vanishing of the Lie derivative of the Ricci tensor along the soliton vector field. In the Kähler case, this vanishing follows from constant scalar curvature, the closedness of the Ricci form, and the classical real holomorphicity of the soliton vector field.

Let $G \rightrightarrows G^{(0)}$ be a Lie groupoid, $\mathrm{A} G \rightarrow G^{(0)}$ its Lie algebroid and $X_1, \dots, X_r$ a family of sections of $\mathrm{A} G$ satisfying a Lie bracket generating condition of Hörmander type. We aim to build a pseudodifferential calculus allowing to study a Helffer-Nourrigat's conjecture on the groupoid $G$; in particular, we want differential operators of the form $\sum_{i = 1}^r X_i^2$ to have an invertible symbol. In this article we achieve the geometrical part of this construction by defining a "weighted" version of the deformation to the normal cone $\mathrm{DNC}(G, G^{(0)}) \rightrightarrows G^{(0)} \times \mathbb{R}_+$. Heuristically, we deform $G$ around $G^{(0)}$ with a "zoom" parameter $t \in \mathbb{R}_+$ by stretching $G$ by $t$ in the directions of the sections $X_i$, $t^2$ along $[X_i, X_j]$, $t^3$ along $[X_i, [X_j, X_k]]$ etc. In the case where $G = M \times M$ we recover a construction of Mohsen, and when the structure is equiregular we recover a construction of van Erp-Yuncken.

We study the Steklov problem on non-compact exterior free-boundary minimal surfaces. Boundary values need not determine a unique harmonic extension, so the operator also requires a prescription at infinity. For proper surfaces in $\mathbb{R}^3$ with compact boundary and finitely many regular ends of finite total curvature, we construct a natural class of such prescriptions. Every resulting operator is self-adjoint with compact resolvent; consequently, its spectrum is discrete, bounded below, and tends to $+\infty$. If the coordinate functions have linearly independent boundary traces, the prescription can be chosen so that these traces are eigenfunctions with eigenvalue $-1$.

In this paper, we prove that any compact immersed disk in a half $3$-space that has constant anisotropic mean curvature and satisfies the anisotropic capillary boundary condition is a homothetic Wulff cap.

We introduce a class of generalised Killing spinors, termed affine Killing spinors (AKS), for which the associated endomorphism is a constant linear combination of the Ricci endomorphism and the identity map. We classify Riemannian spin manifolds admitting an AKS under two additional curvature hypotheses: harmonic curvature and local conformal flatness. Furthermore, we characterise Riemannian spin manifolds that admit a non-zero parallel one-form and an AKS. Additionally, we prove that in dimension three every curvature-homogeneous manifold carrying an AKS is locally homogeneous. Finally, we provide a complete classification of three-dimensional Lie groups equipped with a Bianchi metric admitting an invariant AKS.

A regular observed statistical model may converge to a limit in which a previously identifiable signed parameter becomes identifiable only modulo a reflection. We study the local information geometry of this transition. For a twice differentiable Hellinger embedding with an exact limiting reflection, the observed displacement is forced into the two-jet form \(\varepsilon\lambda J_-+\lambda^2J_+/2\), up to higher-order terms. The mixed jet restores the sign away from the symmetric face, whereas the even jet is the first tangent inherited by the quotient. After nuisance elimination, a positive Gram determinant yields a nondegenerate cross-cap two-jet. The associated local asymptotic theory has three regimes governed by \(\tau_n=\sqrt n\,\varepsilon_n^2\): regular signed LAN, a critical curved Gaussian subexperiment, and a quotient regime with the \(n^{-1/4}\) signed scale. We prove that the same parabolic critical experiment persists for predictive likelihoods along a single stationary dependent trajectory. The limiting quotient has a regular Fisher metric in the invariant coordinate, while its pullback degenerates in the signed coordinate. For a solvable CIR–OU benchmark motivated by coherent sea-clutter observations, we derive the quotient Fisher metric and curvature explicitly and show that the curvature is strictly negative. We also determine the restricted holonomy of the full Amari family: \(\operatorname{Hol}_0(\nabla^{(a)})=SO(2)\) for \(a=0\), whereas \(\operatorname{Hol}_0(\nabla^{(a)})=GL^+(2,\mathbb R)\) for \(a\neq0\). The results separate the intrinsic geometry of the limiting quotient from the transverse geometry of its weak unfolding.

Let H^{n+1} be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^{alpha} {nu} of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^{beta}r; we treat both regimes: beta>1+k{alpha} and beta=1+k{alpha}. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_{infty} respectively in two different cases.

Let $(M,g)$ be a closed negatively curved surface. If $g$ is strictly $\tfrac 19$-pinched and has the same unmarked length spectrum as a hyperbolic metric, we show that $g$ is hyperbolic. As a consequence, we show that any hyperbolic metric on a surface admits a $C^2$-neighborhood in the space of metrics in which it is characterized by its unmarked length spectrum, up to isometry.

The first two authors derived the Weitzenböck-type Formula for a CR Dirac operator. In this paper, we apply the above-mentioned formula and follow the process as Riemannian case to derive a Reilly-type inequality. We give two applications for the CR Reilly inequality. One is a lower bound estimate of the p-mass in terms of the first eigenvalue of the modified hypersurface CR Dirac operator. Another one is to solve the CR Dirac equation on a bounded domain with an $S^{1}$-action and an APS-type boundary condition.

We prove sharp metric entropy estimates for spaces of Kähler potentials. In complex dimension $n$, normalized potentials have Kolmogorov entropy of order $\e^{-n}$ in the background $L^1$ metric. On a polarized manifold, a relative-entropy sublevel has the same order in the Mabuchi–Darvas $d_1$ metric, including its full finite-energy closure. The upper bound is $C_X(1+B)^{n+1}\e^{-n}$ for entropy budget $B$. For toric potentials, the sharp exponent is $n/2$.

This paper is the final installment in our series on the geometric theory of complex Monge-Ampere equations. We study singular Kahler metrics on compact normal Kahler spaces with klt singularities whose volume densities lie in $L^p$ for some $p>1$. We show that a lower bound for the Ricci current in the sense of pluripotential theory is equivalent to a synthetic Ricci lower bound in the sense of RCD theory. As a consequence, every Kahler class on a compact normal Kahler space with klt singularities admits an RCD structure, which makes differential geometric analysis available on singular complex spaces. As an application, we show that the fundamental group of a compact Kahler Calabi-Yau space with klt singularities is almost Abelian. We further establish compactness results for singular Kahler-Einstein spaces.

This paper is the continuation of our earlier work on Holder estimates for Kahler potentials on compact normal Kahler spaces. We establish a geometric counterpart for analytic Holder regularity of Kolodziej. If a singular metric $\omega_\phi$ has Holder continuous potentials with respect to a smooth background metric, then its distance function is Holder continuous with respect to a smooth background distance. The result holds on normal Kahler spaces and yields compactness of the metric completion as well as its identification with the underlying complex space if $\omega_\phi$ is a Kahler current. Under this positivity assumption, our estimates establish the Holder equivalence between the intrinsic canonical Kahler metrics and the extrinsic smooth metrics on Kahler-RCD spaces, particularly on smoothable Kahler-Einstein spaces.

Let $X$ be a compact Kahler manifold. The analytic stability theorem of Kolodziej for complex Monge-Ampere equation states that for any Kahler metrics $\omega$ and $\omega'$ in the same cohomology class, if their volume measures are bounded in $L^p(X)$ (for some $p>1$) and close in $L^1(X)$, then their Kahler potentials are close in $L^\infty(X)$. In this paper, we establish the geometric stability for complex Monge-Ampère equations that $L^1$-closeness of volume measures implies $L^\infty$-closeness for the induced distance functions by $\omega$ and $\omega'$. Consequently, we prove that any non-smooth Kahler current with volume measure bounded in $L^p$ (for some $p>1$) and Ricci current bounded below induces a unique metric space, which turns out to be a compact RCD space homeomorphic to $X$ itself.

We prove that a compact immersed disk of constant mean curvature in $\R^3$ whose boundary is mapped diffeomorphically onto a round circle is a planar disk or a spherical cap.

This paper is concerned with the Weyl problem, i.e., the existence of isometric immersions or embeddings of two-spheres into the three-dimensional Euclidean space or general ambient three-manifolds. We establish two results on the degenerate Weyl problem, namely when the Gaussian curvature $K_g$ is only nonnegative rather than strictly positive. First, for a smooth metric $g$ on the two-sphere, if $K_g$ is strictly positive except at finitely many points where the Hessian of $K_g$ is positive definite, then $g$ admits a global $C^{2,1}$-isometric embedding into $\mathbb{R}^3$. It appears to be the first result on the degenerate Weyl problem with purely intrinsic conditions on $g$. Second, for a general simply-connected ambient three-manifold $(\mathcal{M},{\overline{g}})$, if $K_g \geq K_0 \geq {\rm sec}_{\overline{g}}$ for some constant $K_0$, $(K_g-K_0)^{-1/2} \in L^p$ with $p \geq 2$, and a certain uniform pinching condition holds for approximate nondegenerate isometric immersions, then there exists a $W^{3,p}$-isometric immersion. Alongside we also resolve the nondegenerate Weyl problem (i.e., when $K_g>0$) into general simply-connected ambient three-manifolds for $g$, ${\overline{g}} \in C^{2,1}$.

We study complex rank-two pinor fields on the flat Möbius band and the relation between their deck equivariance and Dirac boundary realizations. With explicit Clifford conventions, the two lifts for each Pin type are represented by $R=c\sigma_1$, where $c=\pm1$ for Pin+ and $c=\pm i$ for Pin-. The relevant longitudinal monodromy is $cJ$, with $J\phi(w)=\sigma_1\phi(-w)$, rather than the fiber matrix alone. Its eigenspaces give shifts $0,1/2$ in the positive case and $1/4,3/4$ in the negative case. We identify the resulting geometric Sobolev spaces and construct self-adjoint elliptic realizations of $D=-i(\sigma_1\partial_x+\sigma_2\partial_w)$ with a local boundary condition defined by the longitudinal tangent field. An exact graph estimate gives compact resolvent and Fredholm mappings on nonnegative Sobolev levels. The spectrum consists of a single unpaired transverse branch and paired branches whose longitudinal shifts alternate with transverse parity. Consequently, the kernel and spectral symmetry depend on the lift and boundary condition; the two negative Pin lifts have eta values $1/2$ and $-1/2$ in the stated convention. We also give a secular equation and an exact kernel criterion for the full translation-invariant family of local self-adjoint elliptic boundary lines.

We obtain restrictions on the geometry and topology of minimal equatorial fillings in $\mathbb{S}^3$ of the lowest possible Morse index.

This paper proves generic structural stability for Riemann solutions to $n \times n$ systems of hyperbolic conservation laws in one spatial dimension. Under assumptions of strict hyperbolicity, genuine nonlinearity, and a regular manifold hypothesis on the Rankine-Hugoniot map, we show that for almost every pair of left and right states, any $n$-wave Riemann solution consisting of Lax-admissible shocks and rarefactions is structurally stable under perturbations of the left state, the right state, and the flux function in the $C^2$ topology. The central new idea is sequential transversality, which chains the $n$ waves through intermediate states and transports their tangent contributions to a common reference point via pushforward maps, reducing the structural stability condition to the invertibility of an $n \times n$ transversality matrix. We apply the results to the $p$-system, polydisperse particle-laden thin films, and machine-learned flux approximations.

We study magnetic flows on closed Riemannian surfaces of genus at least two. Let $(M,g)$ be a surface without focal points and let $\Omega$ be a magnetic field. We prove that, above the Mañé critical value, the absence of conjugate points imposes geometric restrictions on the magnetic flow. First, we obtain a bound for the integral of the Lorentz force in terms of the geodesic curvature of the horocycles of the underlying Riemannian metric. As a consequence, a lower bound on the Gaussian curvature yields an explicit bound in terms of the area of the surface. The proofs combine the global geometry of magnetic geodesics with Liouville's formula for the geodesic curvature in the orthogonal coordinates given by the level sets and geodesics of a Busemann function on the universal cover.

We prove the existence, uniqueness, and partial regularity outside a proper analytic subset for the Kähler current solving the $J$-equation at the birational minimal slope. This confirms Datar–Mete–Song's conjecture 1.5 in \cite{DMS26}. We introduce an analytic threshold, and prove its equivalence to the birational threshold introduced by Datar–Mete–Song. One of the key tools is the approximation of subsolution by Bergman's kernels, which is motivated by the work of Demailly on the approximation of plurisubharmonic functions with analytic singularities. As an application of the Bergman kernel approximation, we combine the results of Fang–Ma \cite{FM26} to give an analytic characterization of the $J$-null locus of a semistable pair $(\alpha,\beta)$. This removes the technical assumptions in \cite{L26b}.