Explains the history and challenges of minimal surfaces.
problem Understanding the regularity of minimal surfaces.
method Historical overview and technical analysis.
result Outlines the evolution and current state of minimal surfaces.
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48 results for “minimal surface theory”
Bound critical points for minimal Radó functions.
problem Counting interior critical points for minimal Radó functions.
method Bounding critical points in terms of boundary data and domain Euler characteristic.
result Bound the number of interior critical points.
In this paper we survey recent developments in the classical theory of minimal surfaces in Euclidean spaces which have been obtained as applications of both classical and modern complex analytic methods; in particular, Oka theory, period dominating holomorphic sprays, gluing methods for holomorphic maps, and the Rieman…
The paper develops a theory for free boundary minimal surfaces with genus at least one.
problem Finding minimal surfaces with specific genus and boundary conditions.
method Using sweepouts of surfaces of genus g≥1 and m≥1 ideal boundary components, the paper constructs a min-max theory for free boundary minimal surfaces.
result The width for the area functional can be achieved by a bubble tree limit of branched genus g free boundary minimal surfaces with nodes.
Complex analysis aids in studying minimal surfaces.
problem Understanding minimal surfaces in Euclidean spaces.
method Complex-analytic techniques applied to conformal minimal surfaces.
result New results on approximation, interpolation, and general position properties.
Constructs cmc doublings of minimal surfaces via min-max theory.
problem Construct cmc doublings of minimal surfaces.
method Uses min-max theory and catenoid estimate.
result Constructs ε-cmc doublings of Σ for small ε > 0.
Study compares different complexity criteria for free boundary minimal surfaces.
problem Comparing different complexity criteria for free boundary minimal surfaces.
method Global theory of free boundary minimal surfaces.
result Provides a complete picture of how area, topology, and Morse index compare.
Anisotropic min-max theory constructs stable minimal surfaces in 3-manifolds.
problem Constructing stable anisotropic minimal surfaces in 3-manifolds.
method Anisotropic min-max theory, removable singularity theorems.
result Constructs stable anisotropic minimal surfaces in 3-manifolds without singularities.
The paper studies a new class of affine maximal surfaces with singularities.
problem Understanding the properties of affine maximal surfaces with singularities.
method Defining a new subclass of affine maximal surfaces and applying Euclidean minimal surface theory.
result Affine maxfaces satisfy an Osserman-type inequality and do not contain non-trivial improper affine fronts.
The paper explores stable surfaces in Einstein-Maxwell theory, proving mass bounds and nonexistence results.
problem Exploring stable surfaces in static Einstein-Maxwell space-time.
method Using mean-stable surfaces theory to prove properties of lapse functions and mass bounds.
result Proves ADM mass is bounded by Hawking quasi-local mass.
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.
An article based on a four-lecture introductory minicourse on minimal surface theory given at the 2013 summer program of the Institute for Advanced Study and the Park City Mathematics Institute.
We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of associate minimal vertical graphs. We prove that an associate surface of a vert…
The study finds a way to create minimal surfaces with specific shapes in 3-manifolds.
problem Creating minimal surfaces with prescribed genus in 3-manifolds with positive Ricci curvature.
method Develops a min-max theory and shows deformability of surfaces in a generic metric.
result Establishes a theorem for producing minimal surfaces with prescribed genus.
Proof that stable minimal surfaces in 3D are flat.
problem Classification of stable minimal surfaces in .
method Index theory for Dirac operators on twisted spinor bundles.
result Every complete two-sided stable minimal surface in is flat.
Minimal Surfaces in the Three-Dimensional Sphere and Minimal Hypersurfaces of Type Number Twomath.DG
We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of t…
Geometry of -minimal surfaces in math.DG
Survey on geometric properties of special minimal surfaces.
problem Understanding -minimal surfaces in .
method Systematic geometric study of -minimal surfaces.
result Fundamental results in the theory of -minimal surfaces.
New theorem connects minimal and maximal surfaces, affecting graphness.
problem Understanding graphness of minimal surfaces in different spaces.
method Introducing a new deformation family and proving Krust-type theorems.
result Graphness of minimal surfaces in isotropic 3-space affects deformed surfaces.
We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used …
Minimal volume vector fields on surfaces via calibrations.
problem Finding minimal volume vector fields on Riemann surfaces.
method Theory of calibrations to write the equation of minimal volume vector fields.
result Equation of minimal volume vector fields on Riemann surfaces.
We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called thin position. Thin position is defined usin…
Harmonic maps intersect all minimal surfaces with bounded curvature.
problem Intersection of harmonic maps with minimal surfaces.
method Nonconstant conformal harmonic maps intersecting bounded curvature minimal surfaces.
result Harmonic maps intersect every nonflat properly embedded minimal surface of bounded curvature.
The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in for any . These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…
We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …
New proof of Smale conjecture for RP^3 and lens spaces using min-max theory.
problem Proving the Smale conjecture for specific spaces.
method Minimal surfaces and min-max theory.
result New proof of Smale conjecture for and lens spaces.
Study on metrics of surfaces with curvature and boundary conditions.
problem Understanding metrics of surfaces with curvature and boundary conditions.
method Revisits and generalizes Ricci-Curbastro's theorem to constant mean curvature surfaces and minimal hypersurfaces.
result New versions of Ricci-Curbastro's theorem for surfaces with curvature and boundary conditions.
Survey on real forms of a complex equation and their connection to surface theory.
problem Describing real forms of the complex -Toda equation and their geometric implications.
method Analyzing the integrability of Maurer-Cartan forms for different real forms of loop groups.
result Each real form of corresponds to a specific surface class with integrable frames.
Helicoids and vorticesmath.DG
New minimal surfaces found from vortex crystals.
problem Minimal surfaces and vortex crystals.
method Gluing helicoids into minimal surfaces.
result New minimal surfaces and vortex crystals discovered.
Survey on algebraic minimal cones and nonassociative algebras.
problem Exploring algebraic minimal cones and nonassociative algebras.
method Recollections and recent developments in the field.
result Dedication to Vladimir Miklyukov's memory.
Funções elíticasmath.DG
Karcher reimagined elliptic functions using geometry.
problem Understanding and controlling elliptic functions.
method Geometrical approach to rewrite elliptic function theory.
result Optimal control over elliptic function behavior and image values.
Introduces minimal surfaces to undergraduates.
problem Teaching minimal surfaces to non-specialist undergraduates.
method Elementary calculus of several variables, accessible to second-year undergraduates.
result Minimal surfaces theory accessible to undergraduates.
Unstable minimal surfaces are the unstable stationary points of the Dirichlet-Integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used. The application of this method for minimal surfaces in the Euclidean spacce was presented in \cite{s3}. We…
Minimal and CMC surfaces in can be treated via their associated family of flat $\SL(2,\C)$-connections. In this the paper we parametrize the moduli space of flat $\SL(2,\C)$-connections on the Lawson minimal surface of genus 2 which are equivariant with respect to certain symmetries of Lawson's geometric construc…
Unified approach to totally ramified values in various surface theories.
problem Totally ramified values in value distribution theory, normal family theory, and Gauss maps of surfaces.
method Bloch--Ros principle applied to various surface theories.
result Unified approach to phenomena concerning totally ramified values.
New minimal surfaces in spheres with complex topologies from capillarity.
problem Constructing minimal surfaces in spheres with rich topologies.
method General construction of embedded minimal and constant mean curvature surfaces in using capillary hypersurfaces.
result Non-trivial sphere bundles over various base spaces, including Stiefel manifolds and complex quadrics.
New proof of timelike minimal surfaces using split-harmonic maps.
problem Interpolating a split-Fourier curve to a timelike minimal surface.
method Using split-harmonic maps to solve the singular Björling problem.
result Solved the interpolation problem for timelike minimal surfaces.
Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deform…
We use Colding--Minicozzi lamination theory to study the systole of large genus minimal surfaces in an ambient three-manifold of positive Ricci curvature.
Minimal surfaces help prove a conjecture about special metrics.
problem Proving Arthur L. Besse's conjecture about CPE metrics.
method Using the theory of minimal surfaces.
result The conjecture is proven for 3D manifolds.
A surface M is called p-minimal if one of the coordinate functions is p-harmonic in the inner metric. We show that in the twodimensional case the Gaussian map of such surfaces is quasiconformal. In the case when the surface is a tube we study the geometrical structure of such surfaces. In particularly, we establish the…
In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus . We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …
Minimal hyperbolic surface diameter grows logarithmically with genus.
problem Finding the smallest possible diameter of hyperbolic surfaces.
method Random construction, lattice point counting, and exploration of random trivalent graphs.
result Minimal diameter is asymptotic to log(g) as genus g approaches infinity.
Extends theory of minimal surfaces to elliptic special Weingarten surfaces.
problem Classify surfaces of finite total curvature.
method Extends Jorge-Meeks formula and uses it to classify surfaces.
result Planes are the only elliptic special Weingarten surfaces with total curvature < 4π.
Minimal stretch factor for non-orientable surfaces is small.
problem Finding the minimal stretch factor for pseudo-Anosov homeomorphisms on non-orientable surfaces.
method Adapting Thurston's theory of fibered faces for non-orientable 3-manifolds.
result The minimal stretch factor is asymptotically on the order of 1/g.
We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An …
Definite knots' quotients remain definite via Seifert surfaces.
problem Definiteness of knots and their quotients.
method Equivariant minimal genus Seifert surfaces.
result Quotients of definite periodic knots are definite.
In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of non-compact symmetric spaces. It also includes the theory of constant mean curvature one surfaces in hyperbolic 3-space. Moreover, a Chern-Os…
In this paper, we prove a generalization of Rado's Theorem, a fundamental result of minimal surface theory, which says that minimal surfaces over a convex domain with graphical boundaries must be disks which are themselves graphical. We will show that, for a minimal surface of any genus, whose boundary is "almost graph…