Derives Lagrangian for minimal surfaces, proving tangential variations vanish.
problem Variational calculus for minimal surfaces.
method Lagrangian formulation, pullback covariant derivative, geometric argument.
result Tangential variations vanish for minimal surfaces.
Noether theorem applied to variational problems on hyperbolic surfaces.
problem Variational problems on hyperbolic surfaces.
method Noether's theorem on symmetry and conservation laws.
result Application to geometric problems on hyperbolic surfaces.
Constructs surfaces with conical singularities using variational methods.
problem Creating Hamiltonian Stationary Surfaces with specific singularities.
method Variational methods and convergence process similar to Ginzburg-Landau analysis.
result Obtained surfaces with prescribed conical singularities related to optimal Wente constants.
Paper derives formulas for surface variations in shell theory.
problem Deriving first variation formulas for surfaces in thin shell theory.
method Using strain-displacement relations from thin shell theory.
result Provides formulas for linear Weingarten surfaces as stationary points.
The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator fo…
We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.
Study connects surface classes to conservation laws.
problem Understanding CMC surfaces in space forms.
method Relates moment class to cohomology class, shows variational origin.
result Both classes have a variational origin as Noether currents.
Eight different refinements of trapped surfaces are proposed, of three basic types, each intended as potential stability conditions. Minimal trapped surfaces are strictly minimal with respect to the dual expansion vector. Outer trapped surfaces have positivity of a certain curvature, related to surface gravity. Increas…
Analyzes surfaces minimizing mean curvature variation using PDEs.
problem Finding surfaces of minimum mean curvature variation.
method Develops an analytic theory using partial differential equations.
result Establishes existence and regularity of minimizers.
We study the variational problem for N-parallel curves on a Finslerian surface by means of Exterior Differential Systems using Griffiths' method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler-Lagrange equations for this type of variational problem…
Through using the semidiameter (in connection to: the mean radius and surface radius) of a convex closed hypersurface in Rn≥2 as an sharp upper bound of the variational (1,n)∋p-capacity radius, this paper settles a restriction/variant of S.-T. Yau's \cite[Problem 59]{Yau} from the surface area to t…
We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived …
Survey on discrete minimal surfaces and their properties.
problem Discretizing minimal surfaces in Euclidean space.
method Polyhedral surfaces with parallel face offsets and circle patterns.
result All simply connected discrete minimal surfaces can be constructed from circle patterns.
Variational autoencoders help estimate missing volatility data.
problem Estimating missing points on partially observed volatility surfaces.
method Derive latent variables, construct synthetic surfaces fitting available data.
result Synthetic volatility surfaces can be used for stress testing and exotic option valuation.
A triangulated piecewise-linear minimal surface in Euclidean 3-space defined using a variational characterization is critical for area amongst all continuous piecewise-linear variations with compact support that preserve the simplicial structure. We explicitly construct examples of such surfaces that are embedded and a…
In this paper we present an algorithm to reduce the area of a surface spanned by a finite number of boundary curves by initiating a variational improvement in the surface. The ansatz we suggest consists of original surface plus a variational parameter t multiplying the numerator H0 of mean curvature function def…
Study on conical singularities in 2D surfaces, deriving Polyakov formulas.
problem Analyzing zeta-regularized determinants in surfaces with conical singularities.
method Demonstrated variational and integrated Polyakov formulas for conical singularities, circular sectors, and cones.
result Explicit formulas for the determinant of conical sectors and cones derived.
The paper bounds the index of CMC surfaces with capillary boundary.
problem Bounding the index of CMC surfaces with capillary boundary.
method Comparison of second variations of area and energy, derived second variation formulae.
result The index is bounded linearly by genus, boundary components, and contact angle.
Study connects K3 surfaces to holomorphic metrics, solving complex structure variation.
problem Understanding complex structure variation on K3 surfaces.
method Using Picard-Fuchs equations and lattice polarizations.
result Explicit example of locally conformally flat holomorphic metric.
Using the symmetry group theory of second order PDEs, one finds the symmetry group associated to Tzitzeica surfaces partial differential equation. One studies the inverse problem and one shows that the Tzitzeica surfaces PDE is an Euler-Lagrange equation. One determines the variational symmetry group of the associated …
Minimal surfaces in spheres have unique energy properties.
problem Characterizing minimal surfaces in spheres based on their energy index and eigenvalues.
method Analyzing the second variations of area and energy for minimal immersions.
result New bounds on the energy index and eigenvalues for minimal surfaces in spheres.
Constrained Willmore surfaces are conformal immersions of Riemann surfaces that are critical points of the Willmore energy W=∫H2 under compactly supported infinitesimal conformal variations. Examples include all constant mean curvature surfaces in space forms. In this paper we investigate more generally the crit…
Paper solves curvature assignment on surfaces with sharp points and edges.
problem Prescribing curvatures on surfaces with conical singularities and corners.
method New variational formulation for surfaces with singularities.
result First results for prescribed curvatures on surfaces with singularities.
Bounding geodesic length variation for surface projective structures.
problem Understanding how geodesic lengths change under projective structure variations.
method Bounding the derivative of complex length in terms of the Schwarzian norm.
result Application to cone-manifold deformations of hyperbolic 3-manifolds.
Generates consistent IV surfaces using VAEs and SDE models.
problem Creating arbitrage-free IV surfaces from historical data.
method Combining VAEs with SDE models for parameter distribution, sampling, and decoding.
result Superior out-of-sample performance of the refined VAE model.
Minimal annuli constructed in PSL2 via variational method.
problem Constructing minimal annuli in a non-symmetric 3-manifold.
method Variational method, foliations by minimal surfaces, limit of compact minimal annuli.
result Existence of complete, embedded minimal annuli asymptotic to vertical planes.
The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.
problem Classifying surfaces with constant astigmatism in space forms.
method Classification and construction of surfaces using variational problems and binormal evolution.
result Locally constructed all rotational surfaces of constant astigmatism.
We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, …
Study minimizes CR surfaces in Heisenberg group with rotational symmetry.
problem Minimizing CR surfaces with vanishing CR invariant energy E1 in Heisenberg group. method Proved local uniqueness, classified global surfaces with rotational symmetry, computed second variation.
result Clifford torus is not a local minimizer of E1. Paper derives second variation formula for eigenvalue functionals on surfaces.
problem Determine if a critical metric is a local maximizer for eigenvalue functionals.
method Derive second variation formula for critical metrics and apply to specific cases.
result Flat metric on non-rhombic torus cannot be a conformal maximizer for first eigenvalue.
Minimal surfaces in harmonic conformally flat space are studied.
problem Minimal surfaces in harmonic conformally flat space.
method Variational geometry approach focusing on mean curvature and Willmore functionals.
result Critical points of mean curvature functional are homeomorphic to the sphere.
Derives formulas for determinant of Laplacian on curved surfaces.
problem Calculating the determinant of the Laplacian on higher genus polyhedral surfaces.
method Variational formulas derived with respect to conical points and angles.
result Explicit expression for determinant up to moduli-dependent factor.
We prove by variational means the existence of a complete, properly embedded, genus-one minimal surface in R^3 that is asymptotic to a helicoid at infinity. We also prove existence of surfaces that are asymptotic to a helicoid away from the helicoid's axis, but that have infinitely many handles arranged periodically al…
The paper classifies surfaces with constant skew curvature in 3-space forms.
problem Classifying surfaces with constant skew curvature in 3-space forms.
method Variational characterization and flow of binormal vector field.
result Classification of rotational surfaces with constant skew curvature.
Framework predicts implied volatility surface without arbitrage.
problem Predicting implied volatility surface without static arbitrage.
method Two-step framework: feature selection and deep neural network (DNN) construction.
result DNN model for surface construction removes static arbitrage and reduces prediction error.
Characterizes curves for minimal surfaces in de Sitter space.
problem Minimal surfaces in de Sitter space.
method Variational problem to find critical points of center of mass.
result Curves are critical points of center of mass.
Study on surface configurations with curvature and elasticity.
problem Equilibrium configurations of surfaces with curvature and elasticity.
method Investigates the Euler-Helfrich functional, focusing on axially symmetric surfaces and their variational problems.
result Critical surfaces for the Euler-Helfrich functional, if axially symmetric, satisfy a simpler second order variational problem.
The study classifies surfaces in Berger spheres as Willmore and Hopf tori.
problem Classifying surfaces in Berger spheres as Willmore and Hopf tori.
method Defined a Willmore functional for surfaces in homogeneous spaces and computed its variational formula. Characterized Clifford and Hopf tori as Willmore surfaces satisfying a sharp inequality.
result Clifford and Hopf tori are the only Willmore surfaces in Berger spheres satisfying a specific inequality.
Unified rigidity theorem for cyclic and alternating surfaces.
problem Infinitesimal rigidity of equivariant minimal maps.
method Unified Lie-theoretic framework connecting cyclic surfaces and cyclic harmonic bundles.
result Infinitesimal rigidity for irreducible cyclic surfaces under various variations.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in sub-Lorentzian Heisenberg group.
method First-variation formula derivation and isoperimetric candidates classification.
result Characterization and conjecture of isoperimetric maximizers.
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.
When calculating the index of a minimal surface, the set of smooth functions on a domain with compact support is the standard setting to describe admissible variations. We show that the set of admissible variations can be widened in a geometrically meaningful manner by considering the difference of area functional, lea…
Study the stability of membranes using Helfrich energy and second variation formula.
problem Stability of membranes under various conditions.
method Developed and applied a second variation formula for the Helfrich energy for a class of surfaces.
result Studied the second variation of the area functional for a specific example.
New method calculates cut locus on surfaces without boundary.
problem Computing the cut locus on compact submanifolds.
method Variational convex problem with conic constraints.
result Proven convergence of the approximation method.
Lecture notes on crystallography and discrete surfaces.
problem Mathematical modeling of crystal structures.
method Variational principle and discrete surface theory.
result Most symmetric crystal structures identified.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.