April 2024

Congruence of compact hypersurfaces

This paper proves that two isometric embeddings of a compact hypersurface into a hemisphere of a higher-dimensional sphere are congruent, under certain constraints on their mean curvatures. It does so by employing a generalization of Reilly's formula to space forms.

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Global regularity for 2-varifolds with square-integrable mean curvature

This paper provides criteria ensuring a 2-varifold is induced by a smooth conformal immersion, using density analysis and a recent regularity result. Below energy thresholds, varifolds minimize Willmore energy as in the parametric approach. Examples attaining the thresholds are given.

Maximal extensions and canonical forms for Lorentzian length spaces

This paper suggests a definition of 'Lorentzian space' that allows a functor from Lorentzian manifolds, and studies maximal Cauchy developments and canonical representatives in this context. Key results include: defining Gromov-Hausdorff metrics for noncompact spaces; an explicit non-spacetime maximal development; and 'Hades coordinates' as canonical forms for developme...

Critical Z2 Eigenvalues on the Sphere

This paper studies eigenvalues and eigenfunctions of the Laplace operator on multivalued functions over the complement of 2n points on the sphere. The eigenvalues can be viewed as functions on configuration spaces of points. Critical eigenfunctions serve as local singularity models in gauge theory and are the main focus. The authors prove generic configurations have no ...

March 2024

Spectral gaps on negatively curved surfaces

This paper proves that negatively curved asymptotically hyperbolic surfaces have a uniform lower bound on the imaginary parts of scattering resonances, thus exhibiting an essential spectral gap. This is shown by adapting methods from previous work and answers an open question, without needing the topological pressure condition assumed previously.

Rigidity of Einstein manifolds with positive Yamabe invariant

This paper provides optimal curvature pinching results for closed Einstein manifolds with positive Yamabe invariant in all dimensions. It extends prior results in dimension four, and improves known bounds relating the Yamabe invariant and Weyl tensor for low-dimensional Einstein manifolds. Advances are also discussed regarding an algebraic Weyl tensor inequality in dime...

February 2024

Localizing gravity via optimal shielding

This paper introduces a localized seed-to-solution method to construct solutions to Einstein's equations that exhibit gravity shielding, or localization. The solutions achieve optimal shielding with arbitrarily narrow gluing domains and arbitrarily slow metric/curvature decay, yet enjoy super-harmonic control. New concepts include normalized asymptotic kernel elements, ...

Helicoidal surfaces flowing by mean curvature in H^2 x R

This paper establishes the existence of one-parameter families of helicoidal surfaces in the Riemannian product H^2 x R. These surfaces evolve under mean curvature flow by simultaneously rotating about a vertical axis and translating vertically. The generating curves trace out two infinite properly embedded spiral arms in H^2 centered at a point.

Lagrangian mechanics for continuum body configurations

This paper represents a continuum material body as a 3D differentiable manifold, with the configuration space given by the space of embeddings of this body into R^n. The topology of this infinite-dimensional manifold of embeddings is used to present a first variation formula for Lagrangian mechanics of continuum configurations.

January 2024

Quantization of Maxwell Fields in the Cauchy Radiation Gauge

This paper proves the existence of Hadamard states for Maxwell equations on any globally hyperbolic spacetime by introducing a new 'Cauchy radiation gauge' that suppresses unphysical degrees of freedom. A key ingredient is a new Hodge decomposition for differential forms in Sobolev spaces on complete non-compact Riemannian manifolds.

Classification of helicoidal surfaces by mean curvature

This paper classifies helicoidal surfaces in R3 based on their mean curvature function. A phase space analysis is used to study the surfaces when mean curvature is rotationally symmetric. Results give a classification for even, increasing mean curvature functions. Examples show behaviors when the function vanishes.

Berry phases on Siegel-Jacobi manifolds

The paper analyzes the relationship between Berry phases and connection matrices on the Siegel-Jacobi disk and upper half-plane. It calculates the connection matrix and covariant derivative of one-forms on the extended Siegel-Jacobi upper half-plane. The research connects concepts from physics, like Berry phases and coherent states, to differential geometry of Siegel-Ja...

December 2023

Orbifolds and elliptic curves

We provide a simplified account of constructing the moduli space of elliptic curves as an analytic orbifold, allowing for non-effective group actions. The goal is a self-contained exposition useful for young researchers interested in interactions between differential and algebraic geometry.

November 2023

Recovering metric tensors from boundary measurements

This paper studies the mathematical problem of determining properties of the interior geometry of a manifold from measurements made only on the boundary. The main results provide conditions under which the internal metric tensor, which encodes information about distances and angles, can be stably recovered. Key tools involve spectral analysis and the Dirichlet-to-Neuman...

Lorentzian surfaces with Killing fields

This paper studies Lorentzian surfaces with Killing fields. It defines a map associating vector fields on the circle to conformal classes of such surfaces. This allows characterizing surfaces without conjugate points. The main results classify conformal classes of tori with Killing fields.

Sigma invariants of spheres and projective spaces

This paper determines the sigma invariants of products of spheres and real, complex, and quaternionic projective spaces. It proves these are achieved uniquely by specific metrics. Key techniques involve minimal isometric embeddings, Yamabe metrics, and Simons' gap theorem.

Relations between least area and normal surfaces

This paper introduces a relaxed version of normal surfaces called quasi-normal surfaces. It proves that under certain conditions, every embedded least area surface is quasi-normal with respect to a fine enough triangulation of the 3-manifold. It also shows the intersection patterns of a quasi-normal surface with the triangulation tetrahedra are well-behaved.

Geometric extension of Itô-Wentzell formula for tensor fields advected by stochastic flows

This paper extends the Itô-Wentzell formula to tensor fields advected by stochastic flows of diffeomorphisms on manifolds. It shows how tensor-valued semimartingales transform under pullback/pushforward along such flows, calling it the Kunita-Itô-Wentzell formula. Key results are equations (18),(19) for pullback/pushforward under regularity conditions. This provides a u...

Smoothness of biholomorphisms between families of rigid domains

This paper establishes a smoothness result for families of biholomorphisms between smooth families of strongly pseudoconvex domains with trivial biholomorphism group. It does this by relating the Bergman metrics on the domains and proving a general smoothness result for families of isometries between families of Riemannian manifolds.

Legendrian unknot links and Ginzburg algebras

This paper associates a singular Legendrian unknot link to a quiver and frozen subquiver. It proves the Chekanov-Eliashberg algebra of this Legendrian link is quasi-isomorphic to the relative Ginzburg algebra. As a consequence, the inclusion of the frozen subquiver's Ginzburg algebra into the relative Ginzburg algebra admits a strong relative smooth Calabi-Yau structure.

Nonlinear transformations of the extended BMS symmetry group

This paper derives the nonlinear transformation laws for the leading order metric functions under the extended BMSW symmetry group. This generalizes previous linearized results, providing useful tools for exploring implications of the extended symmetries. A concise summary outlines key results.

October 2023

Modeling vector fields on spheres

This paper introduces new Gaussian process models for learning vector fields defined on the surface of a sphere. The models respect the intrinsic geometry of the sphere, avoiding undesirable artifacts that can occur with simpler extrinsic models. Key innovations include leveraging tools from differential geometry and intrinsic operators like the Hodge star to build suit...

Cortical architecture for visual perception

This paper proposes a mathematical framework using differential geometry to model the functional architecture of the visual cortex. It represents families of cortical cells sensitive to visual features like orientation, scale, curvature etc. as contact and symplectic manifolds. This allows capturing the neural connectivity patterns and receptive field properties that em...

Connected components of residueless meromorphic differentials

This paper classifies connected components of strata of residueless meromorphic differentials on compact Riemann surfaces. It introduces generalized strata, where the residues of poles satisfy certain linear conditions. The connected components of these generalized strata are distinguished by topological invariants called hyperellipticity, spin parity, and rotation numb...

September 2023

Existence of Levi-Civita connections for real calculi

This paper studies the existence of Levi-Civita connections in the context of real calculi, which provide a derivation-based framework for noncommutative Riemannian geometry. The paper focuses especially on real calculi defined over projective modules, and investigates how the existence of a Levi-Civita connection depends on the algebraic structure of the Lie algebra of...

Intrinsic distances as a solution to Weyl's argument against atomistic space

This paper proposes that assigning intrinsic distances locally in atomistic space can solve the issues raised by Weyl's tile argument against atomistic space. The 'mixed account' allows atomistic space to approximate continuous Euclidean space arbitrarily well. It is argued to be no less natural than standard differential geometry and more compatible with a Lewisian ont...

August 2023

A simplified analysis of variational inequalities for obstacle problems

This paper proves that the solution operators of unilateral and bilateral obstacle problems are Newton differentiable, allowing efficient numerical methods. The analysis exploits polyhedricity and projects obstacle problems onto suitable subspaces, avoiding technicalities.

Asymptotic behavior of twisted holomorphic torsion

This paper proves a formula for the leading term in the asymptotic expansion of a modified version of holomorphic analytic torsion, which twists the Dirac operator using a real 3-form. The authors show the leading term is the same as the untwisted case.

Minimal hypersurfaces in the sphere have improved eigenvalue bounds

This paper proves a new lower bound for the first non-zero eigenvalue of the Laplace-Beltrami operator on a closed minimal hypersurface in the sphere. The bound depends explicitly on the dimension and an upper bound for the length of the second fundamental form, improving upon previous bounds that depended only on the dimension. The proof relies on integral estimates in...

Numerical stability of integrators on manifolds

This paper proposes a generalization of nonlinear stability for numerical integrators on Riemannian manifolds. The authors introduce non-expansive systems on manifolds and define B-stability of integrators, inspired by prior work in Euclidean spaces. They prove the geodesic implicit Euler method is B-stable on manifolds with non-positive curvature. Through counterexampl...

Minimal null scrolls in the 3D Heisenberg group

This paper studies timelike minimal surfaces called null scrolls in the 3-dimensional Heisenberg group equipped with a left invariant Lorentzian metric. The main result shows that minimal null scrolls with vanishing multiplication of the coefficient function and its para-complex conjugate in the Abresch-Rosenberg differential can be characterized as surfaces defined by ...

Embedding formalism for anti-de Sitter superspace

This paper introduces a new realization of the extended anti-de Sitter supergroup and uses it to develop a coset construction and differential geometry for anti-de Sitter superspace. The construction leads to an atlas with two charts and chiral transition functions. A manifestly invariant superparticle model is proposed. The most general conformally flat extended superg...

July 2023

Totally geodesic submanifolds of the pseudo-nearly Kähler SL(2,R)×SL(2,R)

This paper studies Lagrangian submanifolds of the pseudo-nearly Kähler SL(2,R)×SL(2,R). It shows they split into four classes based on their behavior with respect to a certain product structure. The paper gives a complete classification of totally geodesic Lagrangian submanifolds of this space.

Cartan calculus on C-infinity ringed spaces

This paper constructs the tangent sheaf, contractions, and Lie derivatives on local C-infinity ringed spaces, proving that the standard Cartan calculus equations hold in this general setting.

Total CR twist of transversal curves in the 3-sphere

This paper investigates the total CR twist functional for transversal curves in the 3-sphere. It addresses the integration of critical curves by quadratures and the existence of closed critical curves. The authors derive Euler-Lagrange equations for critical curves, leading to conservation laws defined by two parameters (the modulus). They introduce concepts like phase/...

May 2023

A gentle introduction to infinite-dimensional quantum mechanics

This paper develops a framework for quantum mechanics on infinite-dimensional spaces. The key contribution is defining a translation-invariant Laplace operator on the infinite-dimensional space R∞ that serves as a quantum Hamiltonian. The operator is shown to be self-adjoint, allowing quantum evolution. An inner product between complex measures equips the space of squar...

February 2023

A glimpse into the profinite world of arithmetic groups

This paper investigates the profinite properties of arithmetic groups, which are fundamental objects linking number theory and geometry. The authors prove that for most high-rank arithmetic groups, there exist non-isomorphic finite index subgroups with isomorphic profinite completions, demonstrating their lack of 'profinite rigidity'. Exceptions are made for arithmetic ...

January 2023

A general audience guide to helix surfaces in anti-de Sitter space

This paper studies helix surfaces in anti-de Sitter space equipped with a family of metrics that deform the standard one. The authors provide explicit descriptions of such surfaces in terms of suitable curves and isometries. The key results are presented accessibly for a general audience.

June 2019

Unraveling the Mysteries of Complex Structures on the Round Sphere

This paper reviews the well-known result that there is no orthogonal complex structure on the round 6-sphere. The arguments rely on the vanishing of the second de Rham cohomology group of S6. Historical context is provided, including the pioneering work of Blanchard which used ideas that anticipated twistor theory.