Papers

We prove that the weighted Laplacian, or equivalently its conjugate Schrödinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity property for complete gradient Ricci shrinkers and apply it to the spectral asymptotics of the weighted Laplacian. We prove that, inside large geodesic balls of radius $R$, the region where the curvature radius is smaller than $R^{-1}$ occupies an asymptotically negligible proportion of the volume. The proof uses the Ricci flow associated with the shrinker, together with the curvature-radius estimates and Sobolev inequalities of Li–Wang.

Let $\X$ be a Hausdorff second-countable smooth orbifold without boundary, possibly noncompact and ineffective. Suppose $\dim\X\leq n$ and $|G_x|\leq B$ for integers $n\geq0$ and $B\geq1$, where $G_x$ is the full stabilizer at $x$. We construct a smooth Hermitian bundle of explicit rank $R(n,B)$ whose fibre at $x$ is a positive multiple of $\C[G_x]$. Adams operations cancel the Bott obstructions on the boundary spheres of a locally finite good triangulation, and the connectivity of Stiefel manifolds gives actual bundles representing the resulting virtual classes. Smoothing preserves all stabilizer representations. Unitary frames give $\X\simeq[M/U(R(n,B))]$ for a smooth manifold $M$ with a proper locally free action; $M$ is compact exactly when $|\X|$ is compact. Thus every compact smooth orbifold is presentable.

We construct explicit Riemannian metrics with positive biorthogonal curvature on $S^2\times T^2$, answering a question of Bettiol. In particular, positive biorthogonal curvature on a closed four-manifold does not force its fundamental group to be virtually free.

The Dinew–Popovici functional is an energy functional for Hermitian symplectic metrics in a fixed Aeppli cohomology class. Its vanishing characterizes the Kähler metrics in that class, providing a variational approach to Kähler geometry. Dinew and Popovici proved that every critical point is Kähler in complex dimension three. We give a negative answer to Erfan Soheil's question about higher dimensions by constructing non-Kähler critical metrics on products of two Kähler surfaces $(S_1,\eta_1)$ and $(S_2,\eta_2)$. These metrics are critical under every variation on the product 4-fold. We characterize the critical product metrics and prove that non-Kähler critical products exist in the Aeppli class $[\eta_1+\eta_2]_A$ precisely when the canonical bundles of the two surface factors are smoothly trivial. As a by-product, we develop a geometric flow driven by the torsion tensor and converging to the fixed Kähler background when this background metric has nonnegative holomorphic bisectional curvature.

In this paper, we study the formation, precise asymptotics, and structural stability of neckpinch singularities in mean curvature flow. Motivated by the static rigidity of cylindrical self-shrinkers established by Colding, Ilmanen and Minicozzi, we first prove a dynamical rigidity result: mean curvature flow of hypersurfaces that are initially graphically close to a generalized cylinder on a sufficiently large scale, subject to a localized quadratic upward bending, inevitably develop a neckpinch singularity in finite time. We also establish sharp asymptotic expansions for the profile functions of these locally evolving graphs. We show that the rescaled graphical radius converges to a specific polynomial profile with quadratic bending, proving that the resulting singularities are nondegenerate. Finally, we establish an openness theorem showing that nondegenerate neckpinches are structurally stable under C^2 perturbations of the initial data. Combined with recent density theorems for the 3-dimensional mean curvature flow by Szekelyhidi, our results confirm that nondegenerate neckpinches constitute a generic and stable phenomenon in 3-dimensional mean curvature flow.

In this paper, we introduce and investigate partially antiinvariant submanifolds of Kenmotsu manifolds. The integrability of the involved distributions is examined, and conditions ensuring that the induced foliations are totally geodesic are obtained. We further explore the geometric properties of these distributions when the associated structure tensors N and P are parallel. By making use of the curvature tensor of Kenmotsu manifolds and the properties of Kenmotsu space forms with constant psi sectional curvature, we derive various geometric characterizations of partially anti-invariant submanifolds. In addition, we characterize totally umbilical partially anti-invariant submanifolds and establish an inequality for partially anti invariant submanifolds of Kenmotsu space forms, including the characterization of the equality case.

Bringing multiscale geometric analysis directly to irregular point clouds remains difficult: quantities such as local dimension, anisotropy, density variation, and geometric transitions are typically estimated through explicit neighborhood, manifold, or graph constructions, or left for neural networks to infer from coordinates. We introduce Heat Field Signatures (HFS), which lift a point cloud to a multiscale family of smooth ambient heat fields, providing a direct interface from discrete samples to geometric analysis. From this field, HFS computes closed-form global and local signatures directly from pairwise distances, capturing heat concentration, intrinsic dimension, anisotropy, and scale transitions. We further introduce the Heat Dimension Spectrum (HDS), a compact summary of multiscale geometric composition. HFS can be used as a closed-form descriptor, a lightweight learned representation, or a geometric feature channel for neural point-cloud models. Across synthetic and real-world benchmarks spanning subcellular, neuronal, tree, and protein data, HFS outperforms strong point-cloud and multiparameter-persistence baselines while substantially reducing end-to-end cost. On SCOP protein-fold classification, HFS improves over the strongest deep baseline by nearly $24$ percentage points using coordinates alone, while standalone HFS representations are exactly rotation-invariant by construction. More broadly, HFS turns a classical heat field into a practical interface for multiscale geometric analysis in modern point-cloud learning.

In this paper we study the global geometry and classification of complete minimal surfaces with embedded ends and finite total curvature in the product space $\mathbb{H}^2 \times \mathbb{R}$. We describe the structure of embedded ends of such surfaces, constraining the combinatorics of their asymptotic polygons at infinity. Focusing on small total curvature values, we establish a complete classification of the possible asymptotic boundaries for embedded minimal surfaces with total curvature $-4\pi$ and $-6\pi$. In the $-4\pi$ case, we prove the surface is either a horizontal catenoid or simply-connected, with asymptotic boundary belonging to one of four explicit configurations. In the $-6\pi$ case, we show that any embedded example must be simply-connected and classify its possible ideal boundaries. We further construct, via an asymptotic Plateau problem solved by area-minimizing surfaces and successive Schwarz reflections, new 1- and 2-parameter families of (possibly non-embedded) simply-connected minimal surfaces with total curvature $-6\pi$ (and, more generally, $-2(2k-1)\pi$) whose ends are embedded and realize asymptotic configurations not previously known to occur.

We study the critical biharmonic equation \[ \Delta^2 u=u^{\frac{n+4}{n-4}}, \] on a complete, connected, and non-compact Riemannian manifold \((M^n,g)\) of dimension \(n\geq 5\) with non-negative Ricci curvature. We establish an optimal pointwise second-order derivative estimate by Bernstein's method and the continuity method. Using invariant tensors, we derive a differential inequality that yields a rigidity result. More precisely, if there exists a positive finite energy solution, then the manifold is isometric to the Euclidean space \(\mathbb{R}^n\), and the solution is given by \[ u(x)= \frac{ \left[\lambda^2 n(n-4)(n^2-4)\right]^{\frac{n-4}{8}} } {\left(1+\lambda |x-x_0|^2\right)^{\frac{n-4}{2}}}, \qquad \lambda>0,\quad x_0\in\mathbb{R}^n . \]

We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_\tau^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of $\mathbb{H}_\tau^d$. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.

We study two means on the five-dimensional Vinberg cone $\Omega$, viewed as a sparse cone of positive definite $3\times3$ matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group $H$ associated with $\Omega$. It is $H$-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining $\Omega$.

Nakamura [J. Comput. Appl. Math. 131 (2001)] proved that the arithmetic–harmonic matrix iteration converges quadratically to the geometric matrix mean, the Riemannian midpoint of the affine-invariant metric on positive-definite matrices. We consider the more general problem of when a Riemannian midpoint is a Gauss compound mean. A Riemannian metric and an affine connection determine three midpoint maps associated with the connection, its metric dual, and the Levi–Civita connection. We show that the Gauss composition of the first two equals the Levi–Civita midpoint if and only if the latter is invariant under one step of the iteration. We give a sufficient condition for this invariance in terms of an isometry which acts as the point reflection about the Levi–Civita midpoint and exchanges the connection with its dual. Under this invariance, the iterations converge quadratically for sufficiently close initial pairs. Nakamura's iteration is an example of this criterion. The criterion also applies to dual pairs of matrix power means whose Gauss composition is the geometric matrix mean. We give a dually flat Hessian example which shows that duality alone does not suffice. We also give non-flat examples with a Euclidean metric and a parallel cubic form in every dimension larger than one. In dimension one, we characterize the invariance completely by using the Matkowski–Sutô equation. For Euclidean dual pairs, we conjecture that the invariance implies that the cubic form is constant. We prove this conjecture in dimension one and for scalar multiples of a constant cubic form.

Xiuxiong Chen proved the existence of $C^{1, \bar 1}$ geodesics in the space of Kähler potentials and the convexity of the $J$-functional along $C^{1, \bar 1}$ geodesics. Analogous results were proved by Collins and Yau for the hypercritical Leung–Yau–Zaslow equation. In this paper, we study a similar picture for generalised Monge–Ampère equations associated with two right-Noetherian polynomials in proper position.

We study Kähler metrics with cone angle $2\pi\beta$ along a smooth divisor, in the range $1/2<\beta<1$. We show that bounded Riemann curvature near the divisor forces the normal bundle sequence to split holomorphically, confirming a condition proposed by Donaldson.

We prove that the gradient graph of any $W^{2,1}$ semi-convex viscosity solution to the three-dimensional subcritical special Lagrangian equation is area-minimizing and smooth. Furthermore, we prove the smoothness of $W^{2,1}\cap C^{1,1/3+}$ semi-convex viscosity solutions. This yields progress on the conjecture proposed by Wang–Yuan [WY13]. The corresponding results in higher dimensions are also established.

We prove Yau's conjecture $\lambda_{1}=2$ for the stacked Clifford tori of Wiygul: for all integers $N\ge2$, $k,\ell\ge1$ and every sufficiently large $m$, every closed embedded minimal surface arising from Wiygul's construction, of genus $k\ell m^{2}(N-1)+1$ in the round three-sphere and resembling $N$ parallel copies of the Clifford torus joined by small catenoidal tunnels, has first Laplace eigenvalue $2$. For $N\ge3$ these surfaces are chains rather than doublings, and the even–odd decomposition on which all previous verifications for gluing constructions rest is not available. The reflection lemma of Choe and Soret reduces the problem to the sector of functions invariant under the symmetry group of the construction, and we show that the lowest nonzero eigenvalue of that sector equals $4+O(m^{-1})$. The value $4$ is the outcome of an exact identity in the limiting weighted graph model: the limiting waist ratios of the construction form the Perron vector of the adjacency operator of the line graph of a path, so that the spectral gap of the path cancels against the total conductance of the tunnels prescribed by the balancing conditions, and what survives is the coefficient of the Jacobi operator of the Clifford torus. The analytic input consists of a conformally invariant channel inequality on a cylinder and of a Poincaré inequality on a periodically perforated torus. The properties of the construction on which the argument rests are isolated in a reduction theorem for closed surfaces decomposed into blocks joined by families of thin channels along the edges of a finite graph, subject to a symmetry assumption and to Poincaré and trace inequalities on the blocks.

We study positive entire solutions of the flat constant $Q/R$-curvature equation in dimension $n\geq5$. We prove that the weak superharmonicity condition $-\Delta u\geq0$ already forces every nonconstant solution to be admissible. With this automatic admissibility, we classify all such solutions as positive constants or standard bubbles. We also show that the weak superharmonicity condition is structurally necessary. Once it is removed, there is a one-parameter family of positive radial entire solutions near every positive constant. These solutions converge to a positive constant at infinity, while both $-\Delta u$ and $-\Delta(u^{(n-2)/(n-4)})$ change sign infinitely many times.

We study connected Riemannian manifolds on which either the sectional curvature or the Ricci tensor is, at each point, strictly positive, strictly negative, or zero, and ask whether the two signs can coexist under pointwise pinching. A general support-rigidity theorem for positive semidefinite divergence-free symmetric tensors is the common analytic mechanism. For sectional curvature in dimension $n\ge 3$, any locally uniform positive lower bound for the absolute pointwise pinching ratio rules out a change of sign. With a fixed pinching constant $\delta_0$, the flat set has no $C^1$ hypersurface piece; it is empty when $\delta_0>1/2$, and is locally porous when $\delta_0=1/2$. These conclusions are sharp in several senses: there are smooth conformally flat local metrics whose sectional curvature changes sign across a flat hypersurface when the pinching degenerates, and there are closed exactly $1/q$-pinched metrics on spheres with isolated flat points. For Ricci curvature, the sharp threshold is \[ \delta_c=\frac1{n-1}. \] A locally uniform gap $\delta_{\mathrm{Ric}}>\delta_c$ forces one Ricci sign globally. Conversely, for every $0<\delta<\delta_c$ there are local sign-changing metrics with exact pointwise pinching $\delta_{\mathrm{Ric}}\equiv\delta$, and exact local examples also exist at $\delta=\delta_c$. Pointwise strictness $\delta_{\mathrm{Ric}}>\delta_c$ is insufficient if the gap collapses at a Ricci-flat interface. For every subcritical $\delta$ we give explicit closed metrics on $\mathbb{S}^1\times\mathbb{S}^{n-1}$, and complete periodic lifts to $\mathbb{R}\times\mathbb{S}^{n-1}$, whose optimal global lower pinching constant is exactly $\delta$. Finally, we prove two ansatz-specific obstructions to compactifying the exact local constructions. The main content of the proof is generated by ChatGPT 5.6 sol and verified by the authors.

Let $(M,J)$ be a compact almost hermitian $4$-manifold with a smooth embedded $J$-holomorphic curve $C$ representing the canonical class. Motivated by the symplectic Bogomolov–Miyaoka–Yau conjecture, we choose a twisted spin$^{\text{c}}$ Dirac operator ${\mathcal D}$ on $M$ satisfying $$ \operatorname{ind}{\mathcal D}=3c_2(M)-c_1^2(M). $$ We then use the curve $C$ to construct a complex-linear perturbation ${\mathcal A}$ of ${\mathcal D}$ whose singular set is $Z_{\alpha}\sqcup C$, where $Z_{\alpha}=\alpha^{-1}(0)$ for a transverse section $\alpha$ of $\Lambda^{1,0}M$ nonvanishing on $C$. Applying Maridakis' index localization theorem, we express $\operatorname{ind}{\mathcal D}$ as the sum of localized contributions: $3c_2(M)$ from $Z_{\alpha}$ and $-c_1^2(M)$ from $C$.

We prove existence and uniqueness for the capillary $L_p$ dual Minkowski problem for $p>q$ and contact angle $\theta\in (0,\frac{\pi}{2})$ in $\mathbb{R}^{n+1}$, $n\geq 3$. We reduce it to a Monge-Ampère type equation with a Robin boundary condition on the unit spherical cap, by building a new auxiliary function to obtain the $C^2$ estimate for $p>q$ in $\mathbb{R}^{n+1}$ ($n\geq 3$), we then prove that there exists a unique smooth solution that solves this problem provided $\theta\in (0, \frac{\pi}{2})$.

In this paper, we study the Kähler–Ricci flow on $\mathbb{C}P^m$-bundles over a product of Kähler–Einstein manifolds, starting from an initial metric with Calabi symmetry. We prove that every finite-time singularity arising along the flow must be of Type~I.

In this paper, we discuss recent developments and open problems concerning the geometry of compact Kähler varieties whose tangent sheaves are pseudo-effective in a strong sense. As our main result, we prove that, after passing to a finite quasi-étale cover, any compact Kähler variety with quotient singularities and pseudo-effective tangent sheaf admits a flat fibration onto a complex torus, with rationally connected fibers. We also obtain an analogous structure theorem for klt compact Kähler varieties, assuming the existence of minimal models in numerical dimension zero; consequently, the result holds unconditionally in the projective setting and for low-dimensional compact Kähler varieties.

We generalize the concept of Bott connection to the de Rham mapping cone situation. As the main results, when the given closed smooth manifold is foliated, we have the mapping cone version of the Bott vanishing theorem and an adiabatic limit of a family of mapping cone connections on the orthogonal complement of the foliation. The second result shows the necessity of a refined metric if we want to see more interactions between geometry and topology in the mapping cone situation.

By using non-abelian Hodge correspondence on compact Sasakian manifolds, we investigate analogous constructions of Hitchin sections of rank $2$ on $3$-dimensional compact Sasakian manifolds. We obtain canonical deformations of $\widetilde{SL_{2}(\R)}$-Sasakian structures giving a diffeomorphism from the vector space of basic quadratic differentials onto a connected component of the space of equivalent classes of sasakian structures of negative basic first Chern classes and topologically trivial CR structures equipped with an appropriate manifold structure.

We improve the Micallef–Moore index estimate for harmonic two-spheres in $n$-manifolds with positive isotropic curvature to the sharp bound $n-3$. Combining with recent work, this completes the diffeomorphism classification of closed manifolds with positive isotropic curvature in the remaining dimensions five and six.

In the present paper, we consider an exclusion process with killing on a proximity graph constructed from a partition of a geodesically complete weighted Riemannian manifold. We rigorously derive its hydrodynamic equation, which is governed by a heat equation with a killing potential term on the manifold. Within our graph discretization framework, we define an empirical density field for the exclusion process and prove its convergence, under an appropriate space-time scaling, to the unique bounded weak solution of the hydrodynamic equation, provided that the weighted manifold is stochastically complete. Combining local regularity for the parabolic equation with techniques from stochastic analysis on manifolds, we prove the uniqueness of the bounded weak solution and give the explicit representation of the solution in terms of the minimal Schrödinger kernel. Our result requires neither a global lower Ricci curvature bound nor global boundedness of the weight function.

We construct a smooth Riemannian metric with strictly positive sectional curvature on \(S^3\times S^3\). Since \(S^3\times S^3\) is even-dimensional and has Euler characteristic zero, our result disproves the positive-curvature case of Hopf's sign conjecture. The metric and the proof were discovered by the Odin Automatic AI Research Agent.

We prove the future-completeness conjecture of Gödeke and Rendall for expanding spatially homogeneous vacuum spacetimes in four spatial dimensions. More generally, an expansion-energy estimate gives future timelike and null geodesic completeness for homogeneous Einstein equations coupled to maps into complete Riemannian targets, through nine spatial dimensions for vanishing potential. In arbitrary dimension we obtain an affine-length criterion which includes positive potential floors and potentials that decay sufficiently slowly along the scalar trajectory. For future-global homogeneous solutions, the critical nine-dimensional bound also holds for a general stress tensor in an explicit normal energy-condition window, including a broad range of perfect fluids. At the opposite time end, we establish a quotient-stable $C^0$-inextendibility criterion for two-step nilpotent cosmologies. Its proof combines boundary localization with an optimal, intrinsically defined allocation of the central correction in timelike homotopies, and continues to apply when the diameter of compact spatial slices collapses. Exact Heisenberg solutions realize both conclusions. On every fixed compact Bianchi~II quotient, the expanding invariant vacuum data split into an open dense set with globally $C^0$-inextendible developments and a codimension-two locally rotationally symmetric locus with analytic compact Cauchy horizons. We determine the chronology-violating region of the exceptional extensions and quantify the exponentially small proper-time scale on which transverse anisotropy replaces the horizon by a singular end.

Let $V$ be a compact connected manifold and $G$ a simply connected solvable Lie group. We study $G$-Lie foliations on $V$ from the point of view of $\mathcal{D}$-module theory, following the approach initiated by Dathe. To each singular foliation $\mathcal{I}$, we associate the $\mathcal{D}_X$-module $\mathcal{M}_{\mathcal{I}} = \mathcal{D}_X / \mathcal{D}_X \cdot \mathcal{I}$ and the derived ring $\mathcal{D}_{\mathcal{I}} := R\mathcal{H}om_{\mathcal{D}_X}(\mathcal{M}_{\mathcal{I}}, \mathcal{M}_{\mathcal{I}})$. We compute the D-irregularity $\mathrm{D\text{-}irr}(\mathcal{I})$ for three classes of solvable Lie foliations: regular homogeneous foliations, the Meigniez foliation with non-polycyclic holonomy group, and an explicit foliation on a compact 5-dimensional manifold. We show that the non-polycyclicity of the holonomy group $\Gamma$ is reflected in the non-vanishing of higher cohomology groups of $\mathcal{D}_{\mathcal{I}}$, establishing a new connection between the geometry of the holonomy group and $\mathcal{D}$-module invariants.

We establish the uniform doubling property for $G$-invariant metrics on a wide class of compact Riemannian homogeneous spaces $(G/K,g)$, describing explicitly the volume growth of the metric balls $B_g(p,r)$. This property was previously known only for specific cases, including abelian Lie groups, the Lie group $SU(2)$ and quotients of $SU(2)\times \mathbb R^n$. Building on the approach of Eldredge, Gordina and Saloff-Coste for $SU(2)$, we refine and develop geometric and Lie-theoretic tools that allow us to replace the explicit identities of the Milnor basis in $SU(2)$ by a general structural assumption on the metric eigenspace decomposition. In particular, we show that the uniform doubling of $(G/K,g)$ is a consequence of the cyclic bracket condition $[\mathfrak{m}_i,\mathfrak{m}_j]=\mathfrak{m}_k$, for $i,j,k$ pairwise distinct, a property that naturally generalizes the Milnor structure of $SU(2)$. We apply our results to $\mathbb Z_2\times \mathbb Z_2$-symmetric spaces, establishing the uniform doubling property for the complete family of $G$-invariant metrics on several classes of generalized Wallach spaces. For compact homogeneous spaces, we also derive a global Poincaré inequality which holds uniformly for the spaces $(G/K,g)$ under consideration, with a constant controlled by the volume doubling constant of the space.