Estimates the first eigenvalue of a Schrödinger operator on closed surfaces.
problem Estimating the first eigenvalue of a Schrödinger operator on closed surfaces.
method Based on Schoen-Yau's work, derives an estimate.
result Derives an estimate of the first eigenvalue.
Study on algebraic surfaces derived from TF-type surfaces using Laplace-Beltrami operators.
problem Analysis of algebraic surfaces from TF-type surfaces.
method Used I and III Laplace-Beltrami operators on TF-type surfaces and applied eliminate methods on software.
result Obtained degrees and classes of algebraic surfaces.
Study describes how operator properties depend on smoothness on surfaces.
problem Understanding operator properties on surfaces with Morse-Smale diffeomorphisms.
method Analyzes pseudodifferential operators and shift operators on closed smooth surfaces.
result Fredholm property of operators depends on Sobolev smoothness exponent.
The study proves a theorem for surfaces using Codazzi operators and investigates parallel mean curvature surfaces.
problem Understanding surfaces with parallel mean curvature in product spaces.
method Intrinsic Klotz-Osserman theorem and Simons' formula.
result The existence of surfaces with parallel mean curvature in product spaces with non-positive Gaussian curvature.
Inverts operator on hyperbolic surfaces, constructing invariant distributions.
problem Constructing explicit inversion formula for X-ray normal operator.
method First, inversion formula for attenuated normal operator on Poincaré disk and closed hyperbolic surfaces. Then, explicit construction of invariant distributions.
result Explicit construction of invariant distributions with prescribed pushforward.
Quantum groups are realized via surface operator junctions in 4d theory.
problem Realizing quantum groups in 4d theory.
method Lifting Wilson lines to surface operators and studying junctions.
result Categorified quantum groups and representations are reproduced.
Introduces Epstein-Poincaré surfaces for G-oper, generalizing classical construction.
problem Generalizing classical Epstein-Poincaré surfaces to complex Lie groups.
method Introduces Epstein-Poincaré surfaces for G-oper, providing a criterion for Anosov holonomy.
result Provides a criterion for the holonomy of G-oper to be Δ-Anosov.
Study eigenvalues and nodal sets of twisted Dirac operators on surfaces.
problem Eigenvalue and nodal set estimates for twisted Dirac operators.
method Derive an inequality relating eigenvalues and nodal sets, using eigenvalue estimates for the Spin^c Dirac operator.
result Eigenvalue estimates for twisted Dirac operators and Liouville type results.
Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.
problem Spectrum of spin Dirac operator on hyperbolic surfaces with pinched geodesics.
method Trace formula for Dirac operator, Huber's theorem, small-time heat trace asymptotic expansion.
result Convergence of Selberg zeta function for degenerating hyperbolic surfaces.
Study Schiffer operators on Riemann surfaces, linking conformal and topological invariants.
problem Investigate Schiffer operators on Riemann surfaces and their connections to conformal and topological invariants.
method Develop calculus for Schiffer and Cauchy operators, derive index theorems, and characterize kernels and images.
result Derive index theorems for Schiffer operators, connecting conformal invariants to topological invariants.
The study finds surfaces with specific curvature properties are essentially known manifolds.
problem Investigating curvature properties on Kähler manifolds.
method Analyzing the curvature operator of the second kind on closed Kähler surfaces.
result Closed Kähler surfaces with six-positive curvature operator of the second kind are biholomorphic to CP2. Study surfaces of finite Chen-type using Beltrami operators.
problem Characterize surfaces with finite Chen-type.
method Analyze Beltrami operators for surface fundamental forms.
result Identify surfaces of finite type in II and III forms.
The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.
problem Investigating spectral properties of the Jacobi operator for surfaces with nonpositive Euler characteristic.
method Proving a sharp upper bound for the second eigenvalue of the Jacobi operator and classifying surfaces attaining this bound.
result Totally geodesic tori maximize the second eigenvalue among compact orientable surfaces with positive genus.
Formula proves Euler characteristic of singularized surfaces.
problem Calculating the Euler characteristic of singularized surfaces.
method Three operations: collapsing, zipping, and double loop identification.
result Formula for Euler characteristic of singularized surfaces.
Constructs spin hyperbolic surfaces with a spectral gap for Dirac operator.
problem Finding spectral gaps for Dirac operators on hyperbolic surfaces.
method Explicit construction of spin hyperbolic surfaces with increasing genus.
result Uniform spectral gap for Dirac operator on constructed surfaces.
The paper explores discrete versions of the Dirac operator on polygonal surfaces.
problem Addressing the immersion problem and conformal invariants on discrete surfaces.
method Develops discrete Dirac operators for polygonal surfaces and studies their properties.
result Preserves key properties of the smooth Dirac operator on discrete surfaces.
Real slices of parabolic opers on Riemann surfaces are studied.
problem Understanding the fixed-point locus of involutions on parabolic opers.
method Investigated the space of parabolic SL(r,C)-opers and their involutions.
result Fixed-point loci of involutions on different descriptions of parabolic opers coincide.
The study improves bounds on surface eigenvalues.
problem Eigenvalue multiplicities on surfaces of various types.
method Generalization from simple cases to surfaces with holes.
result Improved bounds on Laplacian eigenvalues.
Study real slices of SL(r,C)-opers via Riemann surface involution.
problem Understanding geometric properties of real slices of SL(r,C)-opers.
method Action of anti-holomorphic involution σ on Riemann surface X, construction of involution for different descriptions of mSL(r,C)-opers. result Natural parametrization of fixed point locus via differentials on Riemann surface.
Transforms Dirac operators to relate surfaces in 4D.
problem Relating surfaces in 4D via Dirac operators.
method Constructs Moutard transformation for 2D Dirac operators.
result Explicit example creates double points on spectral curve.
Study on Dirac operator spectrum on shrinking surfaces with cusps.
problem Behavior of Dirac operator spectrum on degenerating Riemannian surfaces.
method Adapted pseudodifferential calculus, including Dirac operators and their resolvents.
result Smoothness of spectral projectors and t2logt regularity for the cusp-surgery trace. Study on spectral points of Inoue surfaces with Tricerri metric.
problem Identifying spectral points on Inoue surfaces.
method Analyzing chiral Dirac operators twisted by flat C∗-connections. result No spectral points inside the annulus α−1/4<∣z∣<α1/4, with spectral points on boundary. Study eigenvalues of Dirac operator on surfaces, proving existence and deriving inequalities.
problem Finding optimal bounds for Dirac eigenvalues on spin surfaces.
method Minimization problem within a fixed conformal class, focusing on surfaces.
result Derive isoperimetric inequalities for the Dirac operator on the sphere, complete conformal spectrum characterization.
Proof that stable minimal surfaces in 3D are flat.
problem Classification of stable minimal surfaces in R3. method Index theory for Dirac operators on twisted spinor bundles.
result Every complete two-sided stable minimal surface in R3 is flat. We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delau…
Scattering theory for harmonic one-forms on Riemann surfaces.
problem Understanding harmonic one-forms on Riemann surfaces.
method Constructing scattering theory through boundary value problems and integral operators.
result Explicit expression for the scattering matrix and proof of unitarity.
We study the compact Hermitian spin surfaces with positive conformal scalar curvature on which the first eigenvalue of the Dolbeault operator of the spin structure is the smallest possible. We prove that such a surface is either a ruled surface or a Hopf surface. We give a complete classification of the ruled surfaces …
New method for calculating loop operations on surfaces.
problem Computing algebraic operations on loops in surfaces.
method Using fillings of surfaces by graphs to compute homological intersection number, Lie bracket, and Lie cobracket.
result Effective new approach to standard algebraic operations on loops.
The fermionic signature operator is analyzed on globally hyperbolic Lorentzian surfaces. The connection between the spectrum of the fermionic signature operator and geometric properties of the surface is studied. The findings are illustrated by simple examples and counterexamples.
Method finds invariants of surfaces using differential operators.
problem Finding invariants of surfaces under motion groups.
method Existence of commuting system of invariant partial differential operators and finite system of invariants.
result Any invariant of the surface is a function of these invariants and their derivatives.
New structure found in loops on quasi-surfaces.
problem Understanding operations on loops in quasi-surfaces.
method Defined a quasi-Lie bialgebra structure.
result Natural operations on loops form a quasi-Lie bialgebra.
Guichard's transformations generate Voss surfaces from sine-Gordon solutions.
problem Generating Voss surfaces from sine-Gordon solutions.
method Using Guichard transformations and recursion operators for sine-Gordon symmetries.
result Explicit derivation of Voss nets and length of Guichard sequences.
We consider semidensities on a supermanifold E with an odd symplectic structure. We define a new Δ-operator action on semidensities as the proper framework for Batalin-Vilkovisky formalism. We establish relations between semidensities on E and differential forms on Lagrangian surfaces. We apply these results to Batal…
The Gauss map of a special surface is studied, leading to symmetry conclusions.
problem Understanding the Gauss map of free boundary minimal surfaces.
method Analyzing eigenfunctions of the Jacobi-Steklov operator.
result Rotationally symmetric surfaces have components of their Gauss map as eigenfunctions of the Jacobi-Steklov operator.
Formula calculates index for CR operators on surfaces with boundary punctures.
problem Computing the index for Cauchy-Riemann operators on surfaces with boundary punctures.
method Large antilinear deformations method, generalized to punctured surfaces.
result Involves a non-standard weighted count of boundary zeros in the Euler characteristic term.
The study finds asymptotic expressions for opers on Riemann surfaces.
problem Understanding the asymptotics of opers on Riemann surfaces.
method Analyzes the holonomy of opers and constructs maps to symmetric spaces.
result Limits of maps to the symmetric space correspond to sub-buildings in the asymptotic cone.
Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.
problem Scattering theory for harmonic one-forms on Riemann surfaces.
method Construction of scattering theory from boundary value problems involving systems of curves and jump problems. Explicit expression for scattering matrix using Schiffer operators.
result Unitary scattering matrix and general association of polarizing Lagrangian spaces.
Estimates nodal sets of eigenspinors on closed surfaces.
problem Estimating the size of eigenspinor nodal sets.
method Modified Bochner technique applied to Dirac operator and related equations.
result Derived inequality relating nodal sets to eigenvalues.
The paper describes and classifies surfaces in isotropic 3-space.
problem Characterizing surfaces in isotropic 3-space.
method Analyzes Weingarten and linear Weingarten surfaces in \mathbb{I}^{3} using the position vector and Laplace operator.
result Classifies surfaces in \mathbb{I}^{3} satisfying certain equations.
This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in n-dimensional euclidean space $\EE^n$ to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret re…
The paper defines and proves stabilization for 3-manifold decompositions with multibranched surface intersections.
problem Decomposing 3-manifolds with more than 3 handlebodies and multibranched surface intersections.
method Definition and proof of stabilization operations for these decompositions.
result Stable equivalence of handlebody decompositions with multibranched surface intersections.
In this paper, we make some remarks on José Espinar's paper "Finite index operators on surfaces" [\texttt{arXiv:0911.3767}, to appear in Journal of Geometric Analysis (2011)].
The paper studies discrete centroaffine surfaces in 3D space.
problem Understanding centroaffine invariants and convexity of discrete surfaces.
method Developed structure equations and integrable systems for discrete centroaffine surfaces. Calculated centroaffine invariants and analyzed Laplacian operator.
result Obtained centroaffine invariants and studied their implications on surface convexity.
Paper finds unique eigenproperties of Euclidean operators.
problem Understanding eigenproperties of Euclidean operators.
method Identifies a family of differential operators and their eigenproperties.
result Eigenproperties are related to embedded minimal surfaces and Nitsche conjecture.
Study of twisted Kapustin-Witten equations on Riemann surfaces.
problem Dimensional reduction of twisted Kapustin-Witten equations.
method Kobayashi-Hitchin type correspondence between Nahm pole solutions and opers.
result Corroborates prediction of Gaiotto and Witten.
Study of bound states in quantum layers with confining potentials.
problem Investigating bound states in quantum layers with confining potentials.
method Developed a general approach using parallel coordinates based on the surface but outside its cut locus.
result Discrete eigenvalues exist for certain quantum layers with positive total Gauss curvature.
Disk pairings with zero signature are related to topological surfaces.
problem Relating pairings with zero signature to topological surfaces.
method Cut-and-glue operations to transform pairings and prove connectivity.
result All balanced pairings for a fixed n are connected on a surface with any number of boundary components.
The paper proves properties of Bergman spaces and Schiffer operators on Riemann surfaces with quasicircles.
problem Properties of Bergman spaces and Schiffer operators on Riemann surfaces with quasicircles.
method Proof of isomorphism of Schiffer operators and application to Plemelj-Sokhotski isomorphism and jump decomposition.
result Schiffer integral operator is an isomorphism between Bergman spaces on different subsets of a Riemann surface.