Partial boundary regularity for area-minimizing currents at tangential boundary points.
problem Boundary regularity of co-dimension one area-minimizing currents.
method Proof closely follows Hardt and Simon's boundary regularity result.
result Partial regularity of tangent cones, uniqueness of tangent cone.
We give partial boundary regularity for co-dimension one absolutely area-minimizing currents at points where the boundary consists of a sum of C1,α submanifolds, possibly with multiplicity, meeting tangentially, given that the current has a tangent cone supported in a hyperplane with constant orientation vector; t…
Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.
problem Conservation law for vector fields on surfaces with piecewise smooth boundaries.
method Generalization of the Poincaré-Hopf Theorem for real-analytic vector fields on surfaces with piecewise smooth boundaries.
result Conservation law for vector fields on surfaces with piecewise smooth boundaries.
Study area-minimizing currents with specific boundary properties.
problem Understanding the structure of area-minimizing currents with tangentially immersed boundaries.
method Analyzes n-dimensional area-minimizing currents with boundary constraints. result Near the boundary of the current, the structure is either uncontrolled or a union of controlled hypersurfaces.
We introduce and study co-dimension one area-minimizing locally rectifiable currents T with C1,α tangentially immersed boundary: ∂T is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such T is supported in a s…
Proves new fixed point formulae for complex manifolds with boundary.
problem Fixed points on complex manifolds with boundary conditions.
method Logarithmic Lefschetz fixed point formulae, normal rescaling, relative duality.
result Resonant boundary terms record normal contact and tangential multiplicity.
We develop the notion of Brakke flow with free-boundary in a barrier surface. Unlike the classical free-boundary mean curvature flow, the free-boundary Brakke flow must "pop" upon tangential contact with the barrier. We prove a compactness theorem for free-boundary Brakke flows, define a Gaussian monotonicity formula v…
The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
problem Extending classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.
method Investigated the restricted mean-value property on Riemannian manifolds, focusing on non-tangential boundary behavior.
result Extended a classical result of Fenton to non-positively curved Harmonic manifolds of purely exponential volume growth.
This paper solves a Calderón problem for Beltrami fields on manifolds.
problem Reconstructing a 3D manifold from boundary measurements of Beltrami fields.
method Defined a normal-to-tangential map for Beltrami fields and used it to reconstruct the manifold.
result A real-analytic 3-manifold can be reconstructed from its normal-to-tangential map.
Reconstructing 3D manifolds from boundary electromagnetic data.
problem Reconstructing a compact, connected, real-analytic Riemannian 3-manifold from tangential electric and magnetic fields on its boundary.
method Factorizing Maxwell's equations and using an isometric transform to reconstruct the metric.
result The electromagnetic Dirichlet-to-Neumann map uniquely determines all derivatives of electromagnetic parameters on the boundary.
Sharp stability estimate for tensor tomography in non-positive curvature.
problem Stability estimate for tensor tomography on manifolds with non-positive curvature.
method Pestov identity with localized frequency boundary term.
result Stability estimate of the form L2↦HT1/2. Proposes a new method for better explaining neural network decisions.
problem Challenges in explaining neural network decisions due to base-point choice.
method Introduces tangentially aligned integrated gradients to maximize explanation tangential alignment.
result Optimal base-point maximizes explanation tangential alignment, leading to more accurate interpretations.
Classical Ljusternik-Schnirelmann category is upper bounded by the number of critical points of any bounded from below differentiable functions of Palais-Smale type. Here we achieve an adaptation of this result for the tangential category of foliations. We introduce a weaker type of Palais-Smale function, obtaining a s…
New conditions ensure points can be uniquely represented by combinations of variety elements.
problem Ensuring points can be uniquely represented by combinations of variety elements.
method Conditions on contact locus of general linear spaces.
result Conditions ensuring non tangential weak defectiveness of projective varieties.
Unified method visualizes curvature on curves and surfaces.
problem Visualizing curvature on planar curves and surfaces of revolution.
method Tangential angle parameterization.
result Clear, consistent visualizations without arbitrary parameter tuning.
The notion of Lagrangian H-umbilical submanifolds was introduced by B. Y. Chen in 1997, and these submanifolds have appeared in several important problems in the study of Lagrangian submanifolds from the Riemannian geometric point of view. Recently, the author introduced the notion of tangentially biharmonic submanif…
Study proves rigidity of critical points in hydrophobic capillary systems.
problem Rigidity of critical points in hydrophobic capillary systems.
method Proves rigidity among sets of finite perimeter in the half space, extending to full hydrophobic regime.
result Rigidity of critical points proven in hydrophobic capillary systems.
Harmonic functions on one side of a quasicircle on a compact Riemann surface can be uniquely extended to the other side.
problem Transmission of harmonic functions across a quasicircle boundary on compact Riemann surfaces.
method Analyzing the properties of quasicircles and applying them to harmonic function boundary values.
result A unique harmonic function can be defined on the other side of a quasicircle boundary, preserving boundary values.
Homological stability proved for handlebody mapping class groups.
problem Homological stability for handlebody mapping class groups.
method Categorical framework developed by Randal-Williams and Wahl, allowing for any number of marked discs and boundary points.
result Homology of handlebody groups stabilizes with respect to genus and number of marked discs for all finite degree coefficient systems.
We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…
New definition of envelopes for curves in Lorentz-Minkowski plane, including singular points.
problem Vague classical definitions of envelopes for singular curves in Lorentz-Minkowski plane.
method Introduced a family of smooth curves with singular points, proposed a new definition of envelopes using lightlike tangential data, and examined contact at singular points.
result Properties of envelopes for curves with singular points in Lorentz-Minkowski plane.
We relate the spectral flow to the index for paths of selfadjoint Breuer-Fredholm operators affiliated to a semifinite von Neumann algebra, generalizing results of Robbin-Salamon and Pushnitski. Then we prove the vanishing of the von Neumann spectral flow for the tangential signature operator of a foliated manifold whe…
Paper introduces a new Poisson kernel for strongly pseudoconvex domains.
problem Developing a new mathematical tool for strongly pseudoconvex domains.
method Introducing a maximal plurisubharmonic function called the pluricomplex Poisson kernel.
result The pluricomplex Poisson kernel shares properties with the classical Poisson kernel and reproduces pluriharmonic functions.
Paper finds obstructions to compact Clifford-Klein forms for tangential symmetric spaces.
problem Existence of compact Clifford-Klein forms for tangential symmetric spaces.
method Analyzes tangential homogeneous spaces and provides necessary conditions for their existence.
result New tangential symmetric spaces are found that do not admit compact Clifford-Klein forms.
Study on a new class of meanders with tangential intersections.
problem Enumerating and understanding meanders with transverse intersections.
method Developed a combinatorial framework, identified connections with other objects, and enumerated specific families.
result Completely enumerated several families of singular meanders.
The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.
problem Analyzing boundary conditions and thin-shell limits for viscous operators on arbitrary smooth hypersurfaces.
method Decomposing the ambient Bochner Laplacian into intrinsic and radial pieces, proving results for stress-free and Hodge boundary conditions.
result Universal thin-shell limits for viscous operators on arbitrary smooth hypersurfaces, including stress-free and Hodge boundary conditions.
The paper proves a weak equivalence for topological cobordism categories.
problem Understanding the homotopy type of topological cobordism categories.
method Using smoothing theory and excision in tangential structure, reducing to smooth cobordism categories.
result The classifying space of the cobordism category is weakly equivalent to a specific Thom spectrum.
The paper studies rational higher tangential structures from Lie groups, revealing systematic effects of variations.
problem Understanding higher tangential structures at the rational level and their connections to geometry.
method Defining and studying Spin-Fivebrane and Spin-Ninebrane structures, analyzing their spaces and gauge groups.
result Variations of higher structures systematically involve lower ones, with interesting effects revealed.
Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We gi…
We study the local differential geometry of varieties Xn⊂CPn+a with degenerate secant and tangential varieties. We show that the second fundamental form of a smooth variety with degenerate tangential variety is subject to certain rank restrictions. The rank restrictions imply a slightly refined v…
The Law of Vector Fields is a term coined by Gottlieb for a relative Poincaré-Hopf theorem. It was first proved by Morse and expresses the Euler characteristic of a manifold with boundary in terms of the indices of a generic vector field and the inner part of its tangential projection on the boundary. We give two diffe…
The study characterizes rectifying curves in n-dimensional space.
problem Understanding rectifying curves in arbitrary dimensions.
method Characterization through various conditions and constructions.
result Different ways to characterize rectifying curves in n-dimensional Euclidean space.
We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tange…
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the loc…
Study on properties of tangential hypersurfaces in product-like manifolds.
problem Investigating properties of tangential hypersurfaces in product-like manifolds.
method Analyzing basic properties and computing curvature tensor relations.
result Computed relations involving the Riemannian curvature tensor of tangential hypersurfaces.
The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.
problem Proving cohomology vanishing for free boundary f-minimal submanifolds in Gaussian-weighted Euclidean balls. method The proof uses a weighted Hardy inequality, cancellation in the weighted Weitzenböck curvature operator, and a boundary reduction.
result The space of tangential f-harmonic p-forms vanishes, leading to Hp(M;R)=0. An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space PN(C). By means of the foca…
Let c be a periodic Reeb orbit on the boundary S of a compact star-shaped domain C in R4. We show that if there is an immersed symplectic disc f in C with boundary c then the self-linking number lk(c) of c equals 2 tan(f)-1 where tan(f) is the tangential self-intersection number of f. We also show that if C is convex a…
Investigates polar tangential angles of curves and their monotonicity.
problem Behavior of polar tangential angles of plane curves.
method Proof of monotonicity for certain curves of monotone curvature.
result Nonexistence results for an obstacle problem involving free elasticae.
Revises Gauss's Lemma using metrical distortion and differential slip.
problem Revising Gauss's Lemma in Riemannian geometry.
method Defining metrical distortion and differential slip, showing their geometric implications.
result Geodesically radial volume and length preservation properties.
Study examines tangential real hypersurfaces on Hermite-like manifolds.
problem Characterizing real hypersurfaces on Hermite-like manifolds.
method Introduced tangential real hypersurfaces and derived main identities.
result Discussed contact metric structures in K-contact and cosymplectic cases.
A submanifold is said to be tangentially biharmonic if the bitension field of the isometric immersion that defines the submanifold has vanishing tangential component. The purpose of this paper is to prove that a surface in Euclidean 3-space has tangentially biharmonic normal bundle if and only if it is either minimal…
We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. More precisely we will trace the patterns of inf…
Characterizes convergence of orbits of holomorphic semigroups in the unit disc.
problem Understanding the convergence behavior of orbits of holomorphic semigroups.
method Analyzes the shape of the starlike domain image of the Koenigs function and uses properties of quasi-geodesics.
result Characterizes convergence types (non-tangential and tangential) in terms of the domain's shape and quasi-geodesic properties.
The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.
problem Understanding exceptional sets for radial limits of superharmonic functions on curved manifolds.
method Analysis of radial geodesic rays, Poisson integrals, Green potentials, and Riesz decomposition.
result Sharp bounds on Hausdorff dimensions of exceptional sets for superharmonic functions.
New manifold construction yields Baire-1 functions as cohomotopy groups.
problem Understanding Baire-1 functions on metric spaces.
method Constructing a manifold and analyzing its cohomotopy groups.
result First cohomotopy group of a constructed manifold is the additive group of integer-valued Baire-1 functions.
For a manifold with boundary, the restriction of Chern's transgression form of the Euler curvature form over the boundary is closed. Its cohomology class is called the secondary Chern-Euler class and used by Sha to formulate a relative Poincaré-Hopf theorem, under the condition that the metric on the manifold is locall…
Survey on geodesic flows in 2-surfaces with changing signatures.
problem Generic singularities of geodesic flows in pseudo-Riemannian metrics.
method Analyzing the behavior of geodesics over different domains and near signature-changing curves.
result Geodesics have limited admissible directions near signature-changing points.