arXane › Mathematics › Mathematics
Wednesday, 4 November 2026: 16 new papers, 6 cross-lists and 18 replacements in Differential Geometry (math.DG), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We study uniqueness of scalar-flat K\"ahler metrics in a fixed K\"ahler class on an asymptotically locally Euclidean (ALE) K\"ahler manifold, assuming only differentiated $o(1)$ decay to a fixed Euclidean metric on the end. In complex dimension $n\geq2$, under suitable polynomial decay of the reference metric, we prove that the relative Mabuchi functional is finite, continuous, and convex along weak geodesics whose endpoint potentials satisfy $u=O(r^{-\gamma})$, $\gamma>2n-4$, together with the corresponding derivative estimates. Combining this convexity with the analysis of the Lichnerowicz operator in weighted spaces, we obtain uniqueness by a perturbation argument. Decay improvement and comparison of the asymptotic charts produce a global potential with the required decay for the difference of the two cohomologous scalar-flat K\"ahler forms. The argument also extends to asymptotically conical K\"ahler metrics approaching a fixed Ricci-flat K\"ahler cone under prescribed decay assumptions.
We show that the Belinski-Zakharov soliton construction factorizes into signed Pfaffian tau-functions, revealing a pure-spinor description of the metric and its regularity conditions. We derive analytic restrictions on the integral rod data, prove uniqueness for the asymptotically flat balance problem at fixed rod structure and periods, and give a conditional criterion for global existence. Combining these results with exhaustive enumeration of the admissible data and rigorous existence certificates, we construct a catalogue of regular asymptotically flat solutions representing 356 rod-word classes through twelve turning points, and 52 isolated regular asymptotically locally flat solutions through eight turning points. Analytic positivity and smooth-completion results establish regularity beyond the finite numerical solve. Based on this catalogue, we investigate two organizations of the solution space: generalized rank-two root systems and compositions of simpler endpoint geometries. The endpoint composition rules reproduce exactly all retained asymptotically flat words through twelve turning points.
We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if $(M^n,g(t))_{t\in[0,T]}$ is a complete Ricci flow with nonnegative scalar curvature at $t=0$, the initial metric is parabolic and not Ricci-flat, and the Einstein tensor $\mathrm{Ric}-\tfrac12 Rg$ is bounded above, then parabolicity is preserved along the flow and the conjugate radius of $g(t)$ is finite for every $t\in(0,T]$. As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that $\liminf_{r\to\infty}\Vol(B(p,r))/r^2=0$, has finite conjugate radius.
We prove that there are no proper biharmonic submanifolds with parallel normalized mean curvature vector field in a nonpositive space form, in arbitrary dimension and codimension. We also prove that every biharmonic surface with flat normal bundle in a four-dimensional nonpositive space form is minimal. These results provide substantial support for Chen's conjecture and the generalized Chen's conjecture for hyperbolic space in higher codimensions.
In this note, we begin with considering the simple one-dimensional field theory describing a single free particle on manifold, and apply the BRST formalism and BV quantization to this model following the standard procedures in physics. Through calculations that, while not fully rigorous, have a more transparent physical interpretation, then subsequent geometric identifications, we recover the deformation quantization and algebraic index theorem on symplectic manifold. This note thus elucidates the physical origins and mechanisms of these mathematical results.
This article has two main parts. The first consists of a brief survey of $G_2$-structures with torsion, namely $G_2$-structures admitting a $G_2$-connection with totally skew-symmetric torsion. We focus primarily on the strong condition that the associated torsion $3$-form is closed. We review their basic properties and their connections with the heterotic $G_2$ system, generalized geometry, and six-dimensional geometry. In the second part, we construct two complete cohomogeneity-one strong $G_2T$-structures on $\mathbb{R}\times S^3\times S^3$ with non-zero closed and co-closed characteristic torsion, and non-vanishing characteristic Ricci tensor. Both have an asymptotically cylindrical end; at the other end, the first is asymptotically conical (AC), with a torsion-free $G_2$ cone as its asymptotic model, while the second is asymptotically locally conical (ALC), with a constant-length circle bundle over a Calabi--Yau cone as its asymptotic model. We investigate the induced $SU(3)$-structures on their circle quotients and show that they are globally conformally almost K\"ahler and globally conformally K\"ahler, respectively. To the best of our knowledge, these provide the first complete strong $G_2T$-structures whose underlying metrics are not locally Riemannian products.
Let $(S,V)$ be a marked 2-sphere with $|V|\ge3$, and fix a spherical discrete conformal class. Izmestiev-Prosanov-Wu proved existence and uniqueness for positive prescribed combinatorial curvature in the spherical discrete conformal class on $(S,V)$. In this paper, we identify a region of prescribed combinatorial curvature vectors, allowing some negative components, for which existence still holds.
B\"ar has shown that a fixed boundary metric and a scalar-curvature lower bound do not generally bound total mean curvature from above. We extend this unboundedness to every smooth compact connected manifold of dimension $n\ge3$ with nonempty boundary and every prescribed boundary metric, using scalar-flat metrics. While B\"ar's examples use warped products, our construction uses a nonzero cubic variation of total mean curvature near a flat product, followed by localization and conformal correction. We also construct arbitrarily $C^k$-small perturbations of the flat torical half-cylinder with nonnegative scalar curvature, unchanged boundary metric, an exact product end, and positive total boundary mean curvature.
We present a geometric derivation of the hidden symmetries of the Kepler problem, combining a duality for central-force systems, obtained by interchanging positions and velocities, with a generalized form of Jacobi's theorem. Applied to the dual Kepler problem, this theorem identifies Kepler hodographs of prescribed energy with geodesics of a metric of constant curvature. After adjoining one point at infinity, the resulting geometry is complete. Its Killing vector fields induce a finite-dimensional Lie algebra of infinitesimal symmetries on the corresponding energy hypersurface. The distinguished point at infinity determines a canonical decomposition of the Killing algebra into two vector subspaces, whose associated Noether invariants recover angular momentum and the Runge-Lenz vector.
We study Jack Lee's conjecture that, for a compact strictly pseudoconvex CR manifold $M$, the condition $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{R})$ implies the existence of a pseudo-Einstein structure. For links of isolated singularities, we construct counterexamples in every odd dimension at least three, with $c_1(T^{1,0}M)=0$ in $H^2(M,\mathbb{Z})$ and $b_1(M)=0$. For isolated algebraic singularities of log canonical type and complex dimension at least three, we prove that the real Chern class condition is equivalent to $\mathbb{Q}$-Cartierness of the canonical divisor, log canonicity, and the existence of a pseudo-Einstein structure. Finally, we construct a Du Bois threefold singularity whose link satisfies the real Chern class condition but admits no pseudo-Einstein structure. The initial versions of the proofs were generated with AI assistance and were subsequently revised and verified by the authors.
Yan-Zhu (arXiv:2605.31477) showed the compactness of complete Riemannian manifolds with non-negative Ricci curvature and uniformly convex boundary. In this paper, we give an alternative proof of this compactness theorem.
Let $M$ be a smooth manifold of dimension $n\ge4$, and let $F$ be a smooth, strongly convex Landsberg metric, regular on the entire slit tangent bundle. We prove that $F$ is Berwald if, at each fixed base point, the Minkowski norm is real-analytic on the entire slit tangent space. The proof rests on two analytic rigidity theorems for a single Minkowski space. The first asserts that, in dimension at least $3$, a real-analytic Minkowski norm whose Hessian metric is flat is induced by an inner product. The second asserts that, in dimension at least $4$, the Hessian metric of a real-analytic Minkowski norm is flat if the entire indicatrix carries a nonzero smooth symmetric quartic form satisfying a twisted Codazzi equation. The quartic form need not be real-analytic, and the second theorem is proved independently of the first. Applying these results to the angular Berwald quartic form shows that the metric is Riemannian on the open set of base points on whose fibres the Berwald curvature is not identically zero. Since every Riemannian metric is Berwald, this set must be empty. No analytic structure, compactness or completeness of the base manifold is assumed, and the metric need not be reversible.
Our results are twofold. Inspired by recent work on weighted spheres, we extend the Lai--Wei construction from one to several independent isoparametric inputs, replacing its profile curve by a hypersurface profile to generate minimal hypersurfaces in spheres. Using a curvature criterion, we obtain new infinite families of area-minimizing hypercones from the original construction and its extensions.
K. Broder recently classified the closed Einstein 4-manifolds whose isometry groups have dimension at least three. The proof reduces the problem to the study of cohomogeneity-one Einstein metrics invariant under $SU(2)$, however a few cases are not treated. We reconcile the missing cases using Broder's techniques with minimal modifications.
In this article, we investigate inequalities involving the min-max width associated to the distance function and the area of two-dimensional Riemannian spheres. This geometric invariant naturally arises as min-max methods are applied for the distance function defined on the space of pairs of points. Our methods involve careful constructions of sweepouts that can be performed under the hypothesis that the Riemannian sphere does not contain stable closed geodesics. Improved bounds are obtained under further geometric assumptions (positive curvature, pinching condition, and antipodal symmetry). In the cases of a pinching condition or antipodal symmetry, our inequalities are optimal.
We classify the simply connected toric gravitational instantons whose torus action has four fixed points. They are the hyperk\"ahler multi-Eguchi--Hanson and multi-Taub--NUT metrics with four collinear centres, the asymptotically flat instantons on Li and Sun's manifold $X_3$, up to scale one for each twist at infinity in $(0,1)$ for a fixed orientation of the torus, and, up to scale, a single further instanton with an ALF end, which is not hyperk\"ahler. The last two are Hermitian for neither orientation. In particular, the toric asymptotically flat instanton on $X_3$ with a given twist is unique up to scale, as conjectured by Li and Sun. The proof combines the solitonic form of Geroch's conjecture, which turns smoothness into polynomial equations for the soliton data, with a convexity principle adapted from Aksteiner. At a fixed asymptotic parameter, the cone errors of the conical metrics with a given rod structure are the gradient of a strictly convex function of the rod lengths, and the strict convexity gives uniqueness. Existence follows from an explicit solution for the ALF end and, for the AF ends, from Li and Sun's existence theorem. We also outline how a chart of Aksteiner identifies these AF instantons with those found by Teo.
In this paper, we study the following critical fractional Schr\"odinger equation: \begin{equation} (-\Delta)^s u+V(|y'|,y'')u=K(|y'|,y'')u^{\frac{n+2s}{n-2s}},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{n-3}, \qquad(0.1)\end{equation} where $n\geq 3$, $s\in(0,1)$, $V(|y'|,y'')$ and $K(|y'|,y'')$ are two bounded nonnegative potential functions. Under the conditions that $K(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$, $K(r_0,y_0'')>0$ and $V(r_0,y_0'')>0$, we prove that equation (0.1) has a new type of infinitely many solutions that concentrate at points lying on the top and the bottom of a cylinder. In particular, the bubble solutions can concentrate at a pair of symmetric points with respect to the origin. Our proofs make use of a modified finite-dimensional reduction method and local Pohozaev identities.
Deep neural networks extract powerful latent representations from images, enabling fast and efficient downstream analysis. However, these latent spaces lack explicit geometric structure, leading to possibly poor interpolation fidelity and misleading extrapolation. Latent interpolation techniques assume symmetric transition costs which are often not the case for real world data. Their symmetric nature cannot capture directional processes, such as disease progression, where certain transitions are impossible (e.g. tumor growth dynamics or cognitive improvement in Alzheimer's disease). Although Riemannian metrics extend Euclidean geometry by incorporating curvature, they remain symmetric. We show that understanding asymmetry and not curvature alone directly addresses these problems. To do so we propose a new Finsler-Randers metric learning framework. It incorporates direction-dependent distances, thus enabling the computation of trajectories respecting monotonic sequential constraints. We demonstrate its potential for interpolation and extrapolation on controlled cross-sectional datasets.
We study overdetermined boundary problems for the maximal surface equation with zero Dirichlet data and constant nonzero Neumann data . For the exterior of a bounded domain with connected boundary in $\mathbb{R}^n$, $n\geq2$, we prove that such domain must be the complement of a ball and the solution is radial. In dimension two, we also show that a positive solution on a domain bounded by a properly embedded noncompact connected curve forces the domain to be a half-plane and the solution to be affine. The proofs combine the divergence structure of the equation with the induced geometry of maximal graphs.
In this paper, we construct Feynman propagators on exponentially decaying perturbations of stationary spacetime with compact Cauchy surface. The propagators are defined as parametrices of the wave operator on exponentially weighted Sobolev spaces defined by complex contours $\Gamma$. We then apply this theory to the computation of indexes of Dirac operators using a continuous Wick-like rotation to relate it to the Riemannian case.
For closed 4-manifolds $X_1, X_2$, we give a sufficient condition, in terms of the Seiberg-Witten invariants of $X_i$, for the Dehn twist on $X_1\#X_2$ along the neck to be smoothly nontrivial. This provides a broad generalization of the result of Kronheimer-Mrowka for $K3\#K3$. We also show that the boundary Dehn twists on the punctured connected sums of many pairs of elliptic surfaces are smoothly nontrivial. We also detect new examples of exotic Dehn twists along Seifert fibered 3-manifolds in elliptic surfaces. As a consequence, we exhibit a new phenomenon in which "exotic diffeomorphisms detect exotic smooth structures on 4-manifolds": the triviality of the boundary Dehn twist characterizes the standard nucleus $N(n)$ among the nuclei $N(n)_{p,q}$ homeomorphic to it, provided that $n\equiv0 \bmod 4$. These results are proved by computing the families Bauer-Furuta invariant for a new framing of the families Dirac index.
Let $M$ be a closed star-shaped hypersurface in $\mathbb{R}^{n+1}$ and let $\eta=Hg-h$ be the first Newton transformation of its second fundamental form. For a fixed constant $\alpha>0$ and $2\le k\le n-1$, we study the prescribed sum Hessian curvature equation \[ \sigma_k(\lambda(\eta))+\alpha\sigma_{k-1}(\lambda(\eta))=f(X,\nu). \] Under the natural admissibility condition \[ \lambda(\eta)\in\Gamma_{k-1}\cap \{\lambda:\sigma_k(\lambda)+\alpha\sigma_{k-1}(\lambda)>0\}, \] we establish a global curvature estimate for closed star-shaped solutions with a general positive right-hand side depending on the position vector and the unit normal. A key point is that the sum Hessian operator is nonhomogeneous; the resulting lower-order term in the maximum-principle calculation is controlled by a partial curvature estimate. We also use the augmented-variable identity \[ \sigma_k(\lambda)+\alpha\sigma_{k-1}(\lambda) =\sigma_k(\lambda_1,\ldots,\lambda_n,\alpha) \] to recover the relevant ellipticity and concavity properties. As an application of the a priori estimates, we prove existence and uniqueness for the corresponding prescribed curvature problem under suitable spherical barrier and radial monotonicity assumptions.