Ozawa solution describes surface deformation from Davey-Stewartson II equation.
problem Surface deformation ruled by the Ozawa solution of Davey-Stewartson II equation.
method Soliton deformation of surfaces ruled by the Ozawa solution.
result Explicit singularity of deformed surface at blow-up moment of Ozawa solution.
The paper analyzes self-similar solutions for mean curvature flow in 3D.
problem Analyzing self-similar solutions for mean curvature flow in R3. method Analysis of self-similar solutions for surfaces of revolution, ruled surfaces, and cylindrical surfaces under homothetic helicoidal motions.
result Characterization and explicit families of exact solutions for cylindrical surfaces.
Constructs solutions for gravitational instantons from minimal surfaces.
problem Finding solutions for gravitational instantons.
method Using correspondence between minimal surfaces and gravitational instantons, derived explicit maximal surface solutions.
result Explicit solutions for maximal surface with zero mean curvature.
Researchers found solutions to minimal surface equations in 6D.
problem Equations of minimal surface type in six dimensions.
method Constructed nonlinear entire solutions using parametric elliptic functionals.
result Found solutions to minimal surface equations in 6D.
The paper examines partial regularity of Lipschitz solutions to minimal surface system.
problem Understanding the regularity of solutions to the minimal surface system.
method Investigation of stationary, integral weak, and viscosity solutions; interior gradient estimate using maximum principle.
result Partial regularity results for Lipschitz solutions, including interior gradient estimate.
Study the Ricci flow on Finsler surfaces and find solutions.
problem Existence and uniqueness of solutions to the Ricci flow on Finsler surfaces.
method First, study the Finslerian Ricci-DeTurck flow and find a unique short time solution. Then, use this to find a solution to the original Ricci flow.
result Find solutions to the Ricci flow on Finsler surfaces.
We give a simple proof for the rotational symmetry of ancient solutions of Ricci flow on surfaces. As a consequence we obtain a simple proof of some results of P.Daskalopoulos, R.Hamilton and N.Sesum on the a priori estimates for the ancient solutions of Ricci flow on surfaces. We also give a simple proof for the solut…
Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly elliptic surfaces. We also construct expanding soliton solutions to the flow on the un…
Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the circumference at spatial infinity of Type II ancient solutions on noncompact surfa…
Study classifies ruled surfaces from mean curvature flow solutions.
problem Investigating surfaces with specific mean curvature equations.
method Classifying ruled and translation surfaces in Euclidean space.
result Ruled and translation surfaces are cylindrical.
Complete solutions found for Toda equations on non-compact surfaces.
problem Solving Toda equations on non-compact Riemann surfaces.
method Introduced complete solutions and proved existence and uniqueness using Toda equations and harmonic bundle techniques.
result Existence and uniqueness of complete solutions to Toda equations on non-compact Riemann surfaces.
Novikov equation symmetries, solutions, and pseudo-spherical surfaces studied.
problem Analyzing geometrically integrable Novikov equation properties.
method Lie symmetries, group-invariant solutions, conservation laws, unique continuation, pseudo-spherical surfaces.
result Classification of invariant solutions and existence of analytic metrics for pseudo-spherical surfaces.
Study proves existence of expanding solutions for multiphase surfaces with regular junctions.
problem Existence of self-similar expanding solutions for multiphase surfaces with regular junctions.
method Proves existence of solutions for a multiphase surface with regular junctions using mean curvature flow.
result Multiple self-similar expanding solutions exist for the initial condition of a multiphase surface with regular junctions.
New boundary functions yield infinitely many solutions for minimal surfaces.
problem Existence of multiple solutions for minimal surfaces.
method Lawson-Osserman constructions and Dirichlet problem analysis.
result Infinitely many analytic solutions and at least one nonsmooth solution.
Study finds obstacles to solutions for specific equations on compact surfaces.
problem Existence of solutions to self-dual equations on compact surfaces.
method Depends on Higgs field zeroes and vortex number.
result Infinitely many Higgs fields for which solutions cannot exist.
The paper studies minimal surface flow and translating solitons, proving global solutions and convergence.
problem Finding global solutions and convergence of minimal surface flow and translating solitons.
method Evolved surfaces over convex planar domains evolving by minimal surface flow, proving a priori estimates for translating solitons.
result Global solutions of minimal surface flow converge to translating solitons under suitable conditions.
Weak solutions found for Chern-Ricci flow on complex surfaces.
problem Existence of weak solutions for Chern-Ricci flow on compact complex surfaces.
method Through blow downs of exceptional curves and backwards smooth convergence.
result Existence of weak solutions and smoothing property proved.
Solves a PDE for Landsberg surfaces using new Finsler surface insights.
problem Solving the Landsberg's PDE for Finsler surfaces.
method Reduces the system of non-linear PDEs to a single PDE, the Landsberg's PDE, and solves it.
result Obtains a class of solutions for the Landsberg's PDE.
New solutions found to Ginzburg-Landau equations on surfaces.
problem Existence of novel solutions to Ginzburg-Landau equations on closed surfaces.
method 2D, critically coupled Ginzburg-Landau theory, topology of moduli space.
result Existence of nonminimal, irreducible solutions on nontrivial line bundles.
Study solutions to affine quasi-Einstein equation on homogeneous surfaces.
problem Understanding solutions to affine quasi-Einstein equation on homogeneous surfaces.
method Using the dimension of affine Killing vector fields as a framework, we provide explicit descriptions of solution spaces.
result Very explicit descriptions of solution spaces for gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds.
Sharp bounds found for minimal surface solutions.
problem Finding bounds for minimal surface solutions.
method Analyzing minimal surface equation with specific boundary conditions.
result Sharp bounds established for solutions over certain domains.
Proves existence and uniqueness of solutions for a specific minimal surface equation.
problem Existence and uniqueness of solutions for a singular minimal surface equation.
method Proves existence and uniqueness of classical solutions for the Dirichlet problem.
result Proves existence and uniqueness of solutions for the α-singular minimal surface equation. Study on super-Toda system on Riemann surfaces, analyzing solutions with four types of bubbling.
problem Analyzing solutions to the super-Toda system on Riemann surfaces.
method Blow-up analysis, energy gap results for spinor parts of solutions.
result Showed energy identities for spinor parts of a blow-up sequence of solutions.
Study solutions to metric equations on modified surfaces.
problem Finding constant scalar curvature Kähler metrics on modified surfaces.
method Expanding solutions to extremal metric type equations.
result Developed methods to solve metric equations on modified surfaces.
We establish a correspondence between Darboux's special isothermic surfaces of type (A,0,C,D) and the solutions of the second order PDE : uΔ(u)-|\nabla(u)|^{2}+Φ^{4}=s, s \in R. We then use the classical Darboux transformation for isothermic surfaces to construct a Bäcklund transformation for this equation and prove a …
Study finite curvature solutions on surfaces with nonnegative Gauss curvature.
problem Finite total curvature solutions of Liouville equation on surfaces with nonnegative Gauss curvature.
method Analyzes asymptotic behavior of solutions on complete surfaces.
result Two extremal cases identified: Euclidean plane or flat cylinder, with specific decay conditions.
Study Tikhonov regularization for ill-posed surface equations, analyzing perturbations and applications.
problem Solving ill-posed operator equations with solutions on surfaces.
method Error analysis of Tikhonov regularization considering surface perturbations and vector bundle solutions.
result Error analysis and practical applications demonstrated for functions on surfaces.
Unified study of harmonic maps between pseudo-Riemannian surfaces.
problem Classifying harmonic maps between pseudo-Riemannian surfaces.
method Unified formalism and Bäcklund transformation.
result Unified solutions to harmonic map equations and corresponding maps.
Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.
problem Blowup behavior of regularized solutions to Jang equation inside apparent horizons.
method Two geometric treatments: dilation and translation. Characterization of limits of rescaled and translated solutions.
result Limits of properly rescaled solutions are constant expansion surfaces.
We show that Scherk's first surface, a one-parameter family of solutions to the minimal surface equation, may be written as a linear superposition of other solutions with specific parametric values.
Study finds minimum growth rate for surface solutions.
problem Finding minimum growth rate for surface solutions.
method Analyzes minimal surface equation with zero boundary values over unbounded domains.
result Establishes lower bound for maximum solution values.
Paper constructs counterexamples of higher solutions to self-duality equations.
problem Whether all real sections give rise to global solutions of self-duality equations.
method Using Willmore surfaces and generalized Whitham flow.
result Higher solutions exist on an open dense subset of the Riemann surface, not globally.
Classifies solitons for surface diffusion flow of graphs.
problem Classifying solitons for surface diffusion flow of graphs.
method Classifies solitons including equilibria, self-similar solutions, and travelling waves.
result Classified solitons for surface diffusion flow of entire graphs.
Paper shows non-existence of solutions for minimal surface problems.
problem Non-existence of solutions to Dirichlet problems for minimal surfaces.
method Analyzes minimal graphs of codimension ≥2 over domains with possibly non-C1 boundaries. result Proves non-existence of solutions in various scenarios.
In 2D, a unique solution is proven for a specific equation.
problem Uniqueness of solutions to a specific equation in 2D.
method Analysis on compact Riemannian surfaces without boundary.
result Uniqueness of solutions proven in 2D.
Paper finds flat minimal surfaces in quaternionic projective spaces.
problem Characterizing flat minimal surfaces in quaternionic projective spaces.
method Analyzing totally real minimal surfaces of isotropy order n.
result Linearly full totally real flat minimal surfaces are two surfaces in C^n, one of which is the Clifford solution.
A construction of blowing up solutions to the modified Novikov-Veselov equation is proposed. It is based on the Moutard transformation of two-dimensional Dirac operators and its geometrical interpretation via surface geometry. An explicit example of such a solution constructed by using the Enneper minimal surface is di…
The paper explores connections between three equations via Wick rotations and symmetries.
problem Investigating relations between solutions to specific equations under Wick rotations.
method Analyzing symmetries and transformations of solutions to the minimal surface, zero mean curvature, and Born-Infeld equations.
result Existence conditions and transformations of real and imaginary solutions under Wick rotations.
Global solution found for string heat flow on surfaces.
problem Existence of global weak solutions for bosonic string heat flow.
method Proved existence with smallness conditions on form and potential.
result Global weak solution found, smooth with finitely many singular points.
Proves how to glue solutions on Riemann surfaces with nodes.
problem Solutions of Hitchin's equations on degenerating surfaces.
method Gluing theorem for logarithmic singularities.
result Proves existence of moduli spaces for Higgs bundles.
Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric CPN−1 sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We give a general criterion to construct non-holomorphic…
The paper finds Einstein-Maxwell solutions on Riemann surfaces.
problem Finding Einstein-Maxwell solutions on Riemann surfaces.
method Explicit construction of solutions on S2-bundles over compact Riemann surfaces. result Generalizes existence results and abundance of solutions.
The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form, first introduced by Gill. Building on our previous work, we investigate this flow on complex surfaces. We establish new estimates in the case of finite time non-collapsing, anologous to some known results for the Kahler-Ricci …
In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2 and the associated action functional of this model is finite, then the generali…
Study on combinatorial Yamabe flow on hyperbolic surfaces, proving existence and uniqueness.
problem Existence and uniqueness of solutions to combinatorial Yamabe flow on hyperbolic surfaces.
method Introduced combinatorial Yamabe flow and extended flow with generalized curvature to address potential degeneration of triangles.
result Established existence and uniqueness of solutions to the extended flow under certain conditions.
We address the problem of second order conformal deformation of spacelike surfaces in compactified Minkowski 4-space. We explain the construction of the exterior differential system of conformal deformations and discuss its general and singular solutions. In particular, we show that isothermic surfaces are singular sol…
We consider an ancient solution g(⋅,t) of the Ricci flow on a compact surface that exists for t∈(−∞,T) and becomes spherical at time t=T. We prove that the metric g(⋅,t) is either a family of contracting spheres, which is a type I ancient solution, or a Rosenau solution, which is a type II ancie…
Study of special solutions to surface diffusion flow of ribbons.
problem Investigating the dynamics of special solutions to the surface diffusion flow.
method Complete classification of stationary solutions, qualitative results on shrinkers, translators, and rotators, explicit parametrisation of a shrinking figure eight curve.
result Explicit parametrisation of a shrinking figure eight curve.