Papers

In this paper, we study a one-to-one correspondence between $\epsilon$-unit generalized Killing spinors on a three-dimensional pseudo-Riemannian spin manifold $(M,g)$ and Lagrangian sections of an almost pseudo-Hermitian structure on its principal spin bundle.

We study elastic knots obtained by minimizing the bending energy of a unit loop together with a small multiple of ropelength. We assume a generalized Fáry-Milnor inequality on the $C^1$-boundary of a knot class: every $H^2$-limit curve in the $C^1$-closure has total curvature at least $2\pi$ times the bridge index. Under this assumption we prove that every elastic knot in a one-component BB knot class, that is, a class whose bridge and braid indices coincide, is the round circle covered precisely that many times. The proof does not differentiate the nonsmooth ropelength functional. A minimal closed braid is contracted in the fibres of a solid torus, producing a family of comparison curves with quadratic bending-energy excess and thickness of linear order. This yields a lower thickness bound for the regularized minimizers. Variations on an intermediate scale then give free stationarity of the limiting curve for the normalized bending energy. Constant curvature and free stationarity reduce the classification to a linear distributional ordinary differential equation.

The physical deformation complex of heterotic $G_2$ compactifications couples instanton deformations to geometric fields and their first-order $\alpha'$ corrections. We give an explicit chain-level decoupling of its first-order coefficient complex at the minimal standard embedding, where the gauge bundle is the tangent bundle, the gauge connection is the Levi–Civita connection, and the background $G_2$ structure is torsion-free. The first Atiyah coupling is null-homotopic through a skew covariant derivative extending the induced connection variation. A second homotopy, given by the covariant codifferential of the skew part of an endomorphism-valued cochain, removes the coupling from the bundle sector to the geometric sector. The residual geometric curvature term cancels against the contribution of the first transformation. These identities hold on arbitrary cochains in every canonical degree and yield mutually inverse differential-operator chain maps. Consequently, the cohomology decomposes into instanton and geometric coefficient cohomologies in all degrees. The argument requires only $\mathrm{Hol}(g)\subseteq G_2$. To our knowledge, no previous explicit all-degree splitting removes the internal gauge and geometric couplings of this physically reduced standard-embedding coefficient complex.

We develop a differential-geometric framework for risk-optimal portfolio reallocation. A long-only portfolio is represented as a point of the probability simplex, endowed with a positive-definite Riemannian metric combining market covariance risk with position-dependent concentration risk. The cumulative risk of a rebalancing trajectory is identified with its Riemannian length, so the least-risk transition between a current allocation and a Markowitz target is a geodesic. We compare this route with direct linear rebalancing and projected gradient ascent. When only market covariance risk is priced, the metric is constant and flat, and the geodesic is exactly the straight-line path. Once position-dependent risk is introduced, the geometry becomes curved and the geodesic weakly dominates competing paths with the same endpoints. A Fisher-Rao concentration term produces modest but systematic savings, while an endogenous crowding metric creates non-convex risk ridges that geodesics can bypass through temporary diversification. Numerical experiments, Monte Carlo transitions, and regression analysis show that the largest gains occur when the direct path crosses strongly crowded regions. The framework provides a general geometric formulation of transition management and can accommodate richer risk metrics and transaction-cost structures.

We prove a Michael-Simon inequality for a class of varifolds in Euclidean space that includes all integral varifolds with locally bounded first variation of arbitrary dimension and codimension. In codimension $2$, this inequality is sharp and implies the sharp isoperimetric inequality for varifolds, which attains equality precisely for the round disk. Our proof uses the optimal transportation approach of Brendle-Eichmair.

Given a complete, 3-dimensional, asymptotically flat initial data set for the Einstein-Maxwell equations with vanishing magnetic field, we show that there exists a small universal constant $\mathcal{C}$ such that $m\geq\mathcal{C}(\sqrt{\mathcal{A}/(16\pi)}+\mathcal{Q}^2\sqrt{\pi/\mathcal{A}}\,)$. An analogous statement holds for asymptotically hyperboloidal initial data sets with the ADM mass replaced by hyperbolic energy. Under the additional assumption of axisymmetry, the inequality may be further strengthened to include angular momentum. These results extend the Penrose-type inequalities recently obtained by Allen-Bryden-Kazaras-Khuri to the charged setting.

For every integer $d\ge4$, we prove strong stability for a subfamily of classical rotational hypersurfaces in $\mathbb H^d$ with normalized mean curvature one. The examples are complete, two-sided, properly embedded, and nowhere umbilic, with topology $\mathbb {R}\times\mathbb{S}^{d-2}$. An explicit positive supersolution yields a quantitative stability inequality for all compactly supported test functions. Consequently, endpoint horospherical rigidity fails in every ambient dimension at least four.

LeBrun's Kähler realization $g_m$ of the Taub–NUT metric on $\mathbb{C}^2$ is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple $\alpha g_m$ admits a Kähler immersion into a finite- or infinite-dimensional complex projective space when $m>\alpha/2$, and conjectured that the same holds for every $m>0$. We prove the conjecture. The restriction of the Kähler potential to the axis $z_2=0$ is governed by the Lambert $W$ function, so $\exp(\alpha\Phi_m)$ has a finite radius of convergence as a power series in $|z_1|^2$ although it is real analytic on the whole half-line; the Vivanti–Pringsheim theorem forbids nonnegative Taylor coefficients, and Calabi's criterion fails. We state the mechanism, which Arezzo, Loi, Placini and Zedda recently used for radial metrics, as a general obstruction to Kähler immersions. In statistical terms the axis restriction of $g_m$ would be a natural exponential family with mean domain $(0,\infty)$ and variance function $\mu/(1+2m\mu)$; the argument gives an elementary proof of the known fact, due to Bar-Lev, Bshouty and Enis, that no such family exists with variance function $\mu/(1+c\mu)$ for any $c>0$. The result confirms onemore case of the conjecture of Loi, Salis and Zuddas that Ricci-flat projectively induced Kähler metrics are flat. The analytic core of the proof has been machine-checked in Lean~4.

Let $(Q,g)$ be a Riemannian manifold equipped with a non-identically zero magnetic field represented by a closed $2$-form $\beta$. We allow $Q$ to be non-compact and do not assume that the metric $g$ is complete. We prove that if the magnetic strength, defined as the pointwise norm of $\beta$, attains a strict local maximum on a non-empty compact set $K$, then every sufficiently low energy level carries a contractible periodic magnetic geodesic of the pair $(g,\beta)$ localized near $K$. More precisely, such orbits exist in every neighborhood of $K$, and their lengths converge to zero with the energy. In particular, if $Q$ is compact, then every sufficiently small energy level carries a contractible periodic magnetic geodesic. We also show that, in general, neither the non-emptiness nor the compactness of $K$ can be omitted. Our proof relies on the calculus of variations of the Lagrangian action functional, and uses several new ideas. More precisely, we overcome: (i) the non-exactness of $\beta$ by restricting the minimax to the set of short loops; (ii) the non-completeness of $g$ by combining a compactification of $Q$ with Thom's Jet Transversality and a blow-up for sequences of magnetic geodesics with energy tending to zero; (iii) the possible non-compactness of Palais–Smale sequences by the positivity of the Ricci magnetic curvature for low energy established by the first-named author. Unlike previous work, Struwe's monotonicity argument cannot be used for our purposes, and we rely on a two-Lyapunov-function argument due to Abbondandolo and Majer.

We develop a root-theoretic localization formalism for genus-zero Gromov–Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov–Kontsevich–Pantev–Yu via the matrix of small quantum multiplication by first Chern class. Let $X$ be a Fano fourfold with $h^{3,1}(X)=1$ arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and $X$ is Hodge general, then $X$ is irrational; (b) if every eigenvalue has Jordan defect at most one and $X$ is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented. Using the recently introduced quantum connection approach to irrationality, we obtain from the computations here presented the irrationality of the very general smooth complete intersection of a hyperplane and a quadric in $\operatorname{LG}(3,6)$; the very general smooth quadric section in $G_2/P_2$; and the very general smooth complete intersection of five hyperplanes and a quadric in the spinor 10-fold.

We prove that, on closed three-manifolds, the limit interface of a sequence of stable abelian Higgs critical points with uniformly bounded energy is a finite union of smooth closed immersed geodesics, possibly with multiplicity and self-intersections. This follows from an integral estimate for the diffuse curvature, derived from an integral bound on the discrepancy.

A generalization of the classical Gage's upper bound for the fundamental tone of the Laplacian on complete non-compact Riemannian manifolds to the $p$-Laplacian context is obtained. As an application, upper estimates for the $p$-fundamental tone of complete non-compact submanifolds of space forms with constant nonpositive sectional curvature, and for a class of product spaces, are presented.

We show that a simply connected compact symmetric space of dimension $n\ge 3$ admits a conformal Killing $2$-form which is not Killing if and only if it is homothetic to one of $\mathbb{S}^n$, $\mathbb{CP}^m$ or $\mathbb{HP}^q$.

We give a weak formulation of spectral scalar curvature bounded from below via establishing a dihedral rigidity result. Given some special convex polyhedron, if another metric are of non-negative spectral scalar curvature in the interior, with weighted mean-convex faces, and with its dihedral angles less than or equal to their flat polyhedral model everywhere along the edges, then the metric must be flat. This is motivated by Gromov's definition of a weak notion of non-negative scalar curvature.

We prove that every bounded stable entire solution $v$ of the Allen–Cahn equation in $\R^3$ is one-dimensional. As a consequence, the full De Giorgi conjecture in $\mathbb{R}^4$ is true. We also obtain local curvature estimates for stable solutions. The proof strategy is inspired by the recent breakthrough work of Chan, Fernández-Real, Figalli and Serra [J. Amer. Math. Soc. 2026], by reducing the stabilty condition for the Allen-Cahn equation to a weak stability condition on a surface (the zero set) and then utilizing Gauss-Bonnet formula. For this purpose, we first use the stability condition to get a sublinear bound for a weighted integral that controls the zeros where the solution is far from planar. If such zeros exist, we isolate a bounded set of them and join $1-v^2$ near this set to derivatives of one-dimensional transitions farther away. By controlling the interaction between these transitions, we derive the weak stability condition on the zero set, which is then used to bound a weighted integral of the squared curvature on the regular part of the zero set by a cutoff gradient integral and a controlled error. We use this inequality to bound the intrinsic area and construct logarithmic cutoffs. The resulting compactly supported test function has a negative contribution near this set that exceeds all joining and cutoff errors, contradicting stability.

We prove that, for every positive integer $r$, the positive normalized diameters of isometric quotients of simply connected irreducible compact Riemannian symmetric spaces of rank $r$ have a uniform positive lower bound. The bound is independent of the dimension and of the acting group, which need not be connected. For the higher-dimensional Grassmannian families, we obtain an explicit bound for positive-dimensional acting groups by constructing nonconstant invariant functions of bounded trigonometric degree. The real and complex cases use moment maps. The quaternionic case follows from a quartic rigidity theorem proved by means of invariant four-forms and the classification of compact quaternionic symmetric spaces. We also give examples showing why the rank and the relative scales of product factors must be controlled.

We prove that on a compact locally flat HKT manifold the quaternionic Monge–Ampère equation always has a unique continuous solution. This provides new evidence for the Conjecture of Alesker and Verbitsky.

We first prove that every smooth boundary metric on a compact manifold extends to a metric with any prescribed positive lower bound for Ricci curvature, whereas extensions under stronger positive \(k^{\mathrm{th}}\)-intermediate Ricci curvature conditions may fail due to local obstructions. For negative sectional curvature, we identify a global obstruction to extension. We then consider the class of compact manifolds defined by the existence of a metric with negative sectional curvature and boundary index at most one. On every manifold in this class, we prove that any smooth boundary metric extends to a metric with negative sectional curvature and strictly convex umbilical boundary. We further show that every such initial metric admits a complete asymptotically hyperbolic isometric extension with the same curvature condition and any prescribed conformal infinity. This class of manifolds is closed under boundary connected sums. As a geometric application of these extension results, every smooth metric on \(\mathbb S^n\) admits a strictly convex isometric embedding into \(\mathbb R^{n+1}\) equipped with a complete metric of negative sectional curvature. The proofs combine neck constructions with corner smoothing for upper curvature bounds.

For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(\kappa,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.

The $3$-link snake robot is an example of a non-holonomic mechanical system with rank $2$ distribution in a $5$-dimensional configuration space. It is one of $(2,3,5)$-geometries, and as such, it admits a description by a parabolic geometry of type $(G_2,P)$. Another example of $(2,3,5)$-geometry was well-studied years ago, and it is known that for balls rolling one over the other without slipping or twisting, if the ratio of ball radii is $1:3$, then it is locally isomorphic to the flat model in sense of $(G_2,P)$ parabolic geometries. Answering a question by P. Nurowski, we are looking for parameters of the $3$-link snake robots yielding a locally flat $(2,3,5)$-geometry. We extend the observation of a previous paper that the distributions of the snake robots contain bases generating finite dimensional Lie algebras. We exploit this observation to simplify the exterior calculus of the robots' geometry. This allows us to implement an effective normalization procedure and we obtain an explicit binary quartic invariant of the robot. Finally, we show that it does not vanish for any of the parameters. Therefore, the locally flat model cannot be achieved for these types of snake robots.

We prove linear stability of all steady and expanding gradient Kähler-Ricci solitons. In the expanding case, we prove strict linear stability under very general assumptions. In particular, every asymptotically conical expanding gradient Kähler-Ricci soliton is strictly linearly stable.

Let $(X,d,p)$ be a noncollapsed pointed Gromov–Hausdorff limit of complete $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, where $n\ge4$. We assume that, on each bounded ball, the integral of the $3/2$ power of the small-ball volume deficit relative to the hyperbolic comparison volume is $O(r^3)$ as $r\rightarrow0$. We prove that the metric singular set has Hausdorff dimension at most $n-3$ and sigma-finite $(n-3)$-dimensional Hausdorff measure, thus confirming a particular case of codimension-three regularity conjecture \cite[Conjecture 2.4]{Naber2020}.

A central conjecture in the regularity theory of harmonic maps asserts that the singular set of a stationary harmonic map has codimension at least three. In this paper, we construct counterexamples to this folklore conjecture. We construct stationary harmonic maps whose singular sets $S$ satisfies $\dim S=n-2$ and $\mathcal{H}^{n-2}(S)=0$ . The target is a smooth compact manifold diffeomorphic to $\mathbb{S}^2 \times \mathbb{T}^3$, with metric arbitrarily close in $C^\infty$ to the standard product metric.

We prove that every finite-dimensional Busemann \(G\)-space of dimension at least five has the disjoint disks property (DDP). For a sufficiently small metric sphere \(L=S(c,r)\), we show that every embedded arc contained in an exact distance level is a homotopical \(Z_2\)-set in \(L\). It follows that \(L\) has the disjoint arc-disk property and the disjoint homotopies property. Daverman's product theorem then gives DDP for \(L\times\mathbb R\), and a local avoidance argument at the center yields DDP for the ambient \(G\)-space. Since finite-dimensional Busemann \(G\)-spaces are generalized manifolds, in dimensions at least five the remaining obstruction to the Busemann conjecture is the resolution problem.

In this paper, we realize volumes of representations as real-valued Chern-Simons invariants in Nil, Sol, and Euclidean geometries. To this end, we formulate a Chern-Simons invariant of a pair of connections on a principal bundle that need not be trivial. For a connected closed oriented 3-manifold $M$ and a representation $\rho\colon\pi_1(M)\to G$ into the identity component $G$ of the isometry group of one of these geometries, we construct an auxiliary connection on the associated flat $G$-bundle. We show that, for a suitably normalized invariant polynomial, the Chern-Simons invariant of the auxiliary and flat connections equals the volume of the representation. For the holonomy representation of a geometric structure, this invariant recovers the Riemannian volume. We also compute the Chern-Simons invariant of the Levi-Civita connection for representative closed manifolds in each of these geometries.

Every smooth closed Riemannian surface of genus zero is isometric to the unit sphere carrying a conformal metric, the round metric multiplied by a positive factor. This paper solves the Poisson equation and the eigenvalue problem of the Laplace–Beltrami operator of such a metric with smooth spherical splines, taking the conformal factor as the only description of the geometry. Because the Dirichlet energy is conformally invariant in two dimensions, the stiffness matrix is the round-sphere matrix for every conformal metric, and the factor enters only the load vector, the mean-value constraint and the weighted mass matrix. A surface embedded in space with a conformal parametrization, a surface of revolution, and a surface given only as a triangle mesh are instances of the same problem that differ in how the factor is obtained; the first two are treated here, and the factor is checked against an area identity before any equation is discretized. The unknown is a spline on a fixed spherical triangulation in broken Bernstein–Bézier form, with $C^r$ smoothness imposed algebraically through edge functionals and the conforming space realized by a null-space matrix computed once and reused for every metric. Each step of the computation is stated with the formulas it needs. Experiments on the round metric, a prolate spheroid, a dumbbell of revolution, an Evans–Fung red-blood-cell profile and a prescribed factor with no embedding show the rates $d+1$ in $L^2$ and $d$ in energy for spline degree $d$, the rate $2d$ for eigenvalues with reproduction of the even-order eigenvalues of the round metric up to degree $d$ to roundoff, and, against parametric surface finite elements on the same triangulations, smaller errors at comparable numbers of unknowns for degrees four and six.

In this paper, we shall use the $L^2$ existence theorems for $\partial\bar{\partial}$ provide by Guy Laville to etablish for existence theorem for the $\partial\bar{\partial}$-Neumann operator on any bounded pseudoconvex starred domaine $\Omega$ in $\mathbb{C}^n$ based on the work of D. Spencer on the $\bar{\partial}$-Neumann. As a result, we obtain a new method for resolving the $\partial\bar{\partial}. $

In this paper, we give a complete description for the full isometry groups of a class of right invariant Riemannian metrics on the special unitary group $\mathrm{SU}(2^N)$. These metrics have been used by physicists to study Nielsen's geometric approach for complexities in quantum computations.

We prove linear stability of Kasner spacetimes in the direction of the Big Bang, up to an explicit finite dimensional space of non-decaying self-similar solutions. The fastest growing solution is the linearization of Taub's explicit Bianchi II solution, describing a Kasner transition. All other non-decaying solutions are inhomogeneous analogues of this, and the linearized Kasner metric. The key novelty is a notion of quasinormal modes on Kasner spacetimes, similar to quasinormal modes on stationary black holes spacetimes. All quiescent (as opposed to oscillatory) vacuum Big Bang spacetimes of dimension 4 are asymptotic to a Kasner spacetime from the perspective of a single observer at the Big Bang. They are expected to be highly unstable due to the oscillations conjectured by Belinski, Khalatnikov and Lifschitz. This paper provides a complete description of this instability on a linear level without symmetry assumptions.

Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\geq 1$. For each prescribed volume fraction $s\in(0,1)\setminus\{\frac12\}$, we prove that the isoperimetric region of volume $s \operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exactly two minimizers, a region $E$ and its complement $E^c$. As consequences, for every $m>0$, uniqueness holds for a generic set of metrics $g$ satisfying $m<\operatorname{Vol}_g(M)$, and it also holds for a generic set of pairs $(g,m)$ with $0<m<\operatorname{Vol}_g(M)$. The proof is variational and requires neither boundary regularity nor nondegeneracy of the constrained Jacobi operator. In particular, the results apply even when isoperimetric boundaries are singular.