Papers

Let $M=S^2\times S^2$. We prove that there is a fixed smooth open ball $U\subset M$ such that, for every $\varepsilon>0$, one can find a smooth metric $g_\varepsilon$ and a closed set $S_\varepsilon$ with the following properties: $\sec_{g_\varepsilon}>0$ on $M\setminus S_\varepsilon$; the relative volume of $S_\varepsilon$ is less than $\varepsilon$; every sectional curvature on $U$ is greater than $1$; $-\varepsilon<\sec_{g_\varepsilon}\leq\Lambda$ for one constant $\Lambda$ independent of $\varepsilon$; the volume and diameter have uniform positive lower and upper bounds; and the injectivity radius has a uniform positive lower bound. In fact, $U\cap S_\varepsilon=\varnothing$. The construction starts from the product of the unit round metrics. A small conformal factor, obtained by smoothing the absolute value of the height function on each sphere, has negative definite Hessian away from two thin equatorial bands. It makes every formerly flat mixed plane strictly positive outside a set of arbitrarily small relative volume, while the negative curvature created in the bands is only $O(\varepsilon)$. A second conformal deformation supported in a very small region produces a core on which all sectional curvatures are greater than $1$; its transition has Hessian bounded above by $O(\varepsilon)$. Finally, a diffeomorphism compresses the fixed topological ball $U$ into that core. Pullback preserves all intrinsic bounded-geometry estimates. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.

Let $X$ be a smooth complex projective variety and $D = D_1+\cdots+D_k\subset X$ a divisor with simple normal crossings. Consider Deligne's canonical extension $(F,\nabla)$ of a flat algebraic vector bundle on $X^*:=X\setminus D$ with unipotent monodromy around every component of $D$. We define and compare the various constructions of the extended Chern-Simons classes $$ \mathrm{CS}_p(\nabla^{\mathrm{Del}})\;\in\; H^{2p-1}(X,\mathbb{C}/\mathbb{Z}),\, p\geq 1, $$ attached to $(F,\nabla)$. Our main theorem states that $\mathrm{CS}_p(\nabla^{\mathrm{Del}})$ is torsion in $H^{2p-1}(X,\mathbb{C}/\mathbb{Z})$, for every $p\geq 2$, extending \cite{Reznikov}, \cite{Reznikov2}, \cite{IS-arXiv},\cite{IS-2div}). We treat the case of quasi-unipotent local monodromies via locally abelian parabolic bundles, and deduce the torsion of Chern-Simons classes. We also extend the Deligne-Sullivan theorem \cite{DeSu} on triviality of flat bundle on a finite covering of a smooth manifold, to that of a canonical extension, and provide torsion-bounds on the extended characteristic classes.

Motivated by recent proofs of Gromov's codimension-two volume-growth conjecture, we propose a synthetic theory for scalar lower bound on metric measure Dirichlet spaces.

We prove that any Busemann G-space such that every sufficiently small metric ball is convex is a topological manifold. The key ingredient in the proof is Ivanov's Helly theorem. The appendix contains a counterexample to a question of Berestovskii–Halverson–Repovš.

For origin-symmetric convex bodies in $\R^n$ invariant under $n-2$ orthogonal hyperplanes reflections, we establish the logarithmic Brunn–Minkowski inequality and classify all equality cases without regularity assumptions. We derive the logarithmic Minkowski inequality and rigidity for cone-volume measures, including an explicit classification for bodies of revolution and uniqueness for simplicial polytopes in the prescribed symmetry class.

We describe a version of positive macroscopic scalar curvature motivated by the work of Alpert, Balitskiy, and Guth, and prove that this condition on a manifold implies a bound on its 1-width in terms of its first Betti number. A key tool in the proof is a decomposition of any closed manifold into a family of chains with convenient combinatorial structure, which was inspired by Nabutovsky, Rotman, and Sabourau's work on sweepouts. Finally, using techniques developed by Chodosh, Li, and Liokumovich, we show that for a sufficiently connected manifold, if the universal cover satisfies this curvature condition, then the manifold has a finite cover homotopy equivalent to $S^n$ or connected sums of $S^{n-1} \times S^1$.

We study Ricci solitons on the Riemannian manifold $\mathbb H^2\times\mathbb R$ equipped with the standard metric. A complete classification of soliton vector fields is obtained: they form a four-dimensional affine space, namely a translate of the Killing algebra $\mathfrak{isom}(\mathbb H^2\times\mathbb R)$. All corresponding solitons are expanding. In addition, gradient solitons are fully characterized and shown to form a one-parameter subfamily of the complete family of soliton fields. As a byproduct, every soliton vector field turns out to be affine, preserving the Levi-Civita connection, the curvature tensor, and the Ricci tensor.

In this paper, we study the submanifolds of holomorphic statistical manifolds with vanishing Bochner curvature tensor (Bochner holomorphic statistical manifolds). We study the Lagrangian statistical submanifolds and discuss their conformal flatness under the assumption of vanishing Bochner curvature tensor. Next we study the holomorphic submanifolds of Bochner holomorphic statistical manifolds and prove that the doubly autoparallel holomorphic submanifold of a Bochner holomorphic statistical manifold has vanishing Bochner curvature tensor.

In this paper, we show that the codimension of the singular set of a stable stationary harmonic map to a round $k$-sphere is at least $k+1$ when $k$ is between 3 and 6, and is at least 7 when $k$ is at least 7. The result is sharp in the sense that there exist energy minimizing 0-homogeneous maps in the aforementioned critical dimensions. We also establish non-trivial index bounds for the non-constant harmonic maps from $n$-spheres to $k$-spheres, provided that $n$ is less than $k$.

We solve John Milnor's problem: among all convex regions of a given area on the sphere, the round disk requires the greatest distortion in cartographic projections onto the plane.

Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.

We present a note on two related questions on 4-forms in 8-dimensions. In particular, we clarify that the Cayley form is not unique in determining a metric via the formula of Karigiannis. Additionally, we discuss and prove part of a conjecture of Salamon and Walpuski.

In this note, we prove a priori interior curvature bounds for strictly convex solutions to elliptic Weingarten curvature quotient equations. The proof does not employ the advanced methods involving integral estimates or compactness arguments. Instead, it relies on a concavity inequality for the equation operator and a novel choice of the auxiliary function to carry out an elementary maximum-principle argument. Interior curvature estimates for strictly convex solutions to the special Lagrangian curvature equation in low dimensions also follow as a consequence.

This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.

Let $(M^n, g, f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that (i) $Ric \geq \frac{\nabla_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$; (ii) $(M^n, g, f)$ smoothly converges to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$, we conclude that $(M^n, g, f)$ is isometric to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$. Notably, condition (i) is weaker than the radial flatness condition in \cite{Petersen-Wylie2}.

This paper studies self-shrinkers and the long-time behavior of the inverse $\sigma_k$ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of self-shrinkers determined by their asymptotic data. More precisely, we formulate the self-shrinker equation as a fully nonlinear Dirichlet problem on hyperbolic space with prescribed data on its ideal boundary, and prove that every continuous negative boundary value determines a unique entire spacelike strictly convex self-shrinker. We further study the inverse $\sigma_k$ curvature flow starting from an entire spacelike strictly convex hypersurface satisfying a subsolution condition at infinity and a uniform positive lower bound for its $\sigma_k$ curvature. We prove global existence of the flow and show that the normalized flow converges locally smoothly to the unique self-shrinker with the same prescribed boundary value at infinity.

In this paper, we show that the quantitative data underlying $K$-stability, originally developed in the study of canonical metrics, also detect the existence of positive scalar curvature Kähler metrics in a fixed class.

We introduce the prescribed scalar curvature measure equation on a compact Kähler manifold. For a Kähler class of positive total scalar curvature, we prove that the following are equivalent: the existence of a positive scalar curvature Kähler metric, solvability of this equation for every admissible measure, $d_1$-coercivity of the associated functionals, and uniform geodesic stability along finite-energy $d_1$-geodesic rays. As a consequence, in each fixed Kähler class, the space of positive scalar curvature Kähler metrics is either empty or contractible. We further prove that every Kähler class on a positive-dimensional compact smooth toric Kähler manifold contains a torus-invariant metric of positive scalar curvature. Therefore, for every Kähler class, the prescribed scalar curvature measure equation admits a smooth solution for every admissible measure, unique modulo constants.

We study the compactness question for certain special Lagrangians in semiflat SYZ fibrations (resp. associative submanifolds in Donaldson's proposal of collapsing coassociative K3 fibrations), and give some criterion for when gradient graphs emerge from the adiabatic limit. This gives a partial converse to the Donaldson-Scaduto proposal.

We prove uniform a priori estimates for solutions to degenerate complex Monge–Ampère equations in big cohomology classes, using both auxiliary-function technique developed by Guo, Phong and Tong [On $L^\infty$-estimates for complex Monge-Ampère equations, Ann. of Math. (2) 198 (2023), no.1, 393-418], and quasi-psh envelope approach developed by Guedj and Lu [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, J. Eur. Math. Soc. (JEMS) 27 (2025), no. 3, 1185-1208.]. As an application, we apply our method to prove the Moser-Trudinger and Brezis-Merle-type inequalities for complex Monge-Ampère equations.

We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned landscape. We focus on Calabi-Yau metrics specifically in a non-compact setting with a global potential, so defined geometrically rather than invoking the topological requirements of the Calabi conjecture. In non-compact settings, we can write the metric determinant with respect to a background in terms of a pluriharmonic or real-valued function. Under bounded, nonuniform, and almost low-rank assumptions, we get a partial eigenvalue blow-up effect. In an empirical setting, a Ricci-flat metric will not form, but the blow-up effect is a local condition and can partially hold empirically on open sets. We isolate the Calabi-Yau case in a theoretical setting, and we counteract the corrupted geometries under regularization. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist via geometric analysis, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

In this paper, we prove a gap theorem for $\mathbb{F}$-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an $\varepsilon$-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).

We introduce families of Lagrange functions associated with height and distance-squared functions on implicit plane curves. We show that degeneracies of these families characterize geometric properties of the curves, such as inflections and vertices, and give geometric interpretations of the corresponding Lagrange multipliers in terms of contact with lines and circles.

We study the local metric completion of positive singular conformal metrics near a codimension-two corner when the conformal factor is generated by a compact positive measure on exponent space. The Newton support function determines noncritical weighted-ray accessibility, but at the critical power order support geometry alone is insufficient. We identify the missing invariant as a Laplace transform measuring how exponent mass approaches the critical total-degree face. If the maximal total degree exceeds 2, the local completion fibre over the corner is empty. When all total degrees are at most 2, the fibre is empty or a singleton according as the square root of the critical Laplace profile is nonintegrable or integrable at infinity. We give support-identical measures with opposite critical completion behaviour, sharpness examples, monotone truncation phenomena, intrinsic Lipschitz completion coordinates, and a bi-Lipschitz transfer principle for Newton-comparable densities.

We discuss the advantages of the Calculus of Moving Surfaces over the Euler-Lagrange equation for optimization problems originating in Geometry. An extension of Tensor Calculus, it provides tools for analyzing geometric quantities directly rather than their coordinate representations. This allows us to avoid the many difficulties associated with the use of coordinates, from the untenable complexity of analytical expressions to the virtual impossibility of recovering the geometric interpretation of the final result. As an illustration, we analyze the brachistochrone and give its geometric characterization in terms of its curvature.

We show that quasitoric manifolds can be equipped with an invariant metric for which the torus action is polar. More generally, we show that an infinitesimally polar action of a torus admits an invariant Riemannian metric making it polar if and only if the real orbifold Euler class of its quotient vanishes. Because the orbit spaces of quasitoric manifolds are contractible simple polytopes, this cohomological obstruction trivially vanishes.

We study the extension of the conformal structure of a Riemann surface, obtained as a submanifold of $\mathbf R^n$, across an isolated singular point, under hypotheses that are metric rather than analytic. The main result (1989) is that an isolated singularity is $\textbf{conformally point-like}$ (conformal to a punctured disc) whenever it is $\textbf{$M$-regular}$: the pair (surface minus the point, the point) satisfies a Whitney condition, and the length of the spherical slice $S(0,r)\cap E$ decreases at most linearly in $r$, a condition described as “metric decay”. This is proved via a modulus-of-rings (extremal-length) argument, generalizing the classical planar technique to submanifolds of $\mathbf R^n$. Two classes of examples are treated: surfaces of revolution generated by a single curve, and subanalytic surfaces, for which $M$-regularity is verified directly from Hironaka's structure theory, giving as a corollary that every isolated singularity of an orientable subanalytic surface is conformally point-like. A further original result is a $C^\infty$ counterexample showing that a strict Whitney condition alone does $\textbf{not}$ imply the linear length bound: the two hypotheses in the definition of $M$-regularity are independent.

We introduce a holonomy groupoid for Lie algebroids. This construction generalizes both the holonomy groupoid of a foliation and the adjoint representation of a Lie algebra. We prove that the resulting groupoid is longitudinally smooth and compute its Lie algebroid. When the algebroid of the holonomy groupoid coincides with the given Lie algebroid, the holonomy groupoid is the canonical terminal integration: every source-connected integration admits a unique morphism into it inducing the identity infinitesimally. To construct the groupoid structure on the holonomy groupoid, we utilize the notion of the "flow product" of time-dependent sections of a Lie algebroid. This provides an alternative way to define the groupoid structure for the Weinstein groupoid.

In a parabolic contact geometry a choice of contact form determines a Weyl form consisting of three components: a soldering form, a Weyl connection, and a Rho tensor. An explicit description of the soldering form is well-known. We express the Rho tensor in terms of the Weyl connection in the setting of torsion-free normal parabolic contact geometries. Parabolic contact geometries with a specified contact form have a second canonical connection, the Tanaka-Webster connection. We express the Weyl connection in terms of the Tanaka-Webster connection in the setting of normal parabolic contact geometries. As an application of some of these results we derive formulas for the harmonic curvature components in integrable Legendrian contact geometry, recovering a result of S. Ivanov, D. Vassilev, and S. Zamkovoy.

Statistical watermarks for language models live in the freedom of the signifier: they choose among tokens that are nearly equivalent in meaning, and they are therefore eroded by exactly those transformations which move the form of a text while leaving its content in place. The literature measures such transformations by their endpoint, through the semantic similarity between the original and the rewritten text. We show that the endpoint is the wrong statistic. Adapting the formalism of linguistic loops, we prove that the invariant of a chain of meaning-preserving transformations factorises canonically into an endpoint part and a holonomy in the stabiliser of the initial state, the second of which the semantic deficit cannot see; the loop rotation is parallel transport on the unit sphere of the embedding space, so that the analogy with the Wilson loop becomes a theorem rather than a figure of speech. On the side of the detector we prove an exact identity: the residual statistic is proportional to the number of positions whose seeding window survived intact, from which the decay law $\rho^{h+1}$ follows as the independent-edit corollary. The identity has a disconcerting consequence, which we confirm to three decimal places: at one and the same retention rate the surviving signal may be one half of the original, one quarter of it, or exactly nothing, according only to where the edits fall.