New embedded periodic surfaces constructed in n-dimensional space.
problem Constructing new types of minimal surfaces in higher dimensions.
method Starting with a Jordan curve, a Plateau minimal disk is formed and extended using Schwarz reflections to create an embedded surface.
result Characterization of Jordan curves that generate embedded surfaces in n-dimensional space.
given two minimal surfaces embedded in §3 of genus g we prove the existence of a sequence of non-congruent compact minimal surfaces embedded in §3 of genus g that converges in C2,α to a compact embedded minimal surface provided some conditions are satisfied. These conditions also imply that, if any of th…
Constructs a family of genus three minimal surfaces with parallel ends.
problem Creating embedded doubly periodic minimal surfaces with specific topological properties.
method Constructs a one-parameter family of surfaces with given Weierstrass data and solves the period problem.
result Solves the two dimensional period problem for the constructed surfaces.
Embedded surfaces in a ball have any genus and connected boundary.
problem Existence of embedded free boundary minimal surfaces with specific properties.
method Min-max techniques applied to the unit ball in R3. result Existence of embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
The Weierstrass representation for minimal surfaces in R3 provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more general geometric modeling tool. Minimal surfaces lie in the larger class of harmoni…
The paper proves smooth minimal surfaces in hyperbolic 3-manifolds.
problem Existence of smoothly embedded minimal surfaces in hyperbolic 3-manifolds.
method Analytical proof in hyperbolic geometry.
result Existence of smoothly embedded closed minimal surfaces in hyperbolic 3-manifolds.
Improved bound on first eigenvalue of minimal surfaces in S3.
problem Bounding the first eigenvalue of minimal surfaces in S3. method Proved λ1≥1+εg for embedded minimal surfaces Σ in S3. result Improved bound on the first eigenvalue of λ1. New construction of minimal surfaces in hyperbolic space.
problem Creating minimal surfaces with specific properties in hyperbolic space.
method Presented a new construction method.
result Embedded minimal surfaces with 3 asymptotically totally geodesic ends and arbitrary finite genus.
The paper proves nonexistence and existence results for minimal surfaces in R^4.
problem Proving nonexistence and existence of minimal surfaces in R^4.
method Analyzing complete minimal immersions with finite total curvature and embedded planar ends.
result Existence of embedded minimal spheres in R^4 with 3 embedded planar ends.
New formulas for minimal surfaces with specific end conditions.
problem Existence and explicit formulas for minimal surfaces with embedded planar ends.
method Provided new explicit formulas for genus 0 minimal surfaces in R^3 with 2k+1 embedded planar ends.
result Existence and explicit formulas for minimal surfaces with 2k+1 embedded planar ends for all k ≥ 4.
We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3-manifold N. We also obtain a least area, incompressible, properly embedded, finite topology, 2-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.
Characterizes stable minimal capillary surfaces with specific angles.
problem Understanding stable minimal capillary surfaces with near 0 or π angles. method Curvature estimates for sequences of weakly stable minimal capillary surfaces.
result Characterization of tangential limits of stable minimal capillary surfaces.
Examples of complete minimal surfaces properly embedded in H^2 x R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H^2 x R, not necessarily proper, which are invariant by a vertical tra…
We study properly embedded and immersed p(pseudohermitian)-minimal surfaces in the 3-dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. …
The study finds large area minimal surfaces in certain manifolds.
problem Existence of minimal surfaces with large area in manifolds.
method Analyzes bumpy closed Riemannian n-manifolds for n between 3 and 7.
result Proves the existence of a sequence of minimal surfaces with unbounded area.
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
We prove prove a bridge principle at infinity for area-minimizing surfaces in the hyperbolic space H3, and we use it to prove that any open, connected, orientable surface can be properly embedded in H3 as an area-minimizing surface. Moreover, the embedding can be constructed in such a way that t…
Every point on an asymptotically flat 3D space has a minimal plane nearby.
problem Finding minimal surfaces in asymptotically flat 3D spaces.
method Proving the existence of minimal planes for every point in the manifold.
result Every point in an asymptotically flat 3D space has a complete properly embedded minimal plane.
We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When n=3, this implies apriori area and curvature estimates for these minimal surfaces in terms of the …
In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with nonpositive curvature. We show this result by constructing a non-properly embedded minimal plane in hyperbolic 3-space. Hence, this gives a counterexample to Calabi-Yau conj…
The study establishes curvature estimates and convexity for a specific type of minimal surfaces.
problem Curvature estimates and convexity for a particular class of minimal surfaces.
method Compactness argument and curvature estimates for a family of surfaces.
result Characterization of convexity for properly embedded minimal surfaces with specific curvature conditions.
No minimal charts with exactly seven white vertices found.
problem Finding minimal charts with specific vertex counts.
method Investigating charts representing embedded surfaces in 4-space.
result No minimal chart with exactly seven white vertices exists.
In this paper, we prove that every confomal minimal immersion of an open Riemann surface into Rn for n≥5 can be approximated uniformly on compacts by conformal minimal embeddings. Furthermore, we show that every open Riemann surface carries a proper conformal minimal embedding into R5. One …
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.
Maps on surfaces can be embedded into spheres with minimal dimensions.
problem Embedding periodic maps of surfaces into spheres with the smallest possible dimensions.
method Determining the minimal dimensions m for embeddings of periodic maps of order n on surfaces of genus g into spheres Sm. result For each integer k>1, there exist infinitely many periodic maps such that the smallest possible m is equal to k. In 1997, Collin proved that any properly embedded minimal surface in R3 with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are the only non-planar, non-simply connected, properly embedded, minimal planar domains in $\mathbb…
Researchers find the minimal genus for a specific type of surface bundle.
problem Determining the minimal genus of ΣgimesT2 bundles. method Constructing embedded surfaces to find exact minimal genus values.
result Exact values of minimal genus functions for ΣgimesT2 bundles are determined. Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
The plane and catenoid are the only capillary minimal surfaces outside a unit ball with one end and finite total curvature.
problem Characterizing capillary minimal surfaces in 3D space.
method Analytical proof using geometric properties and curvature constraints.
result The plane and catenoid are the only capillary minimal surfaces under specified conditions.
Every nonflat conformal minimal surface is homotopic to a proper one.
problem Proving homotopy of nonflat conformal minimal surfaces to proper ones.
method Analyzing immersions and fluxes of Riemann surfaces into \(\mathbb{R}^n\) and \(\mathbb{C}^n\).
result Every nonflat conformal minimal immersion is homotopic to a proper one.
Two new triply periodic minimal surfaces of genus 4 discovered.
problem Finding new triply periodic minimal surfaces of genus 4.
method Combination of asymptotic analysis and geometric methods to solve the period problem.
result Two new 1-parameter families of embedded triply periodic minimal surfaces of genus 4.
The study counts minimal surfaces in 3-manifolds with positive Ricci curvature.
problem Counting minimal surfaces in 3-manifolds with positive Ricci curvature.
method An enumerative min-max theorem linking surface counts to topological properties.
result Every 3-sphere of positive Ricci curvature contains at least 4 embedded minimal surfaces of genus 2.
Given a sequence of properly embedded minimal surfaces in a 3-manifold with local bounds on area and genus, we prove subsequential convergence, smooth away from a discrete set, to a smooth embedded limit surface, possibly with multiplicity, and we analyze what happens when one blows up the surfaces near a point where…
In this survey, we discuss various aspects of the minimal surface equation in the three-sphere S^3. After recalling the basic definitions, we describe a family of immersed minimal tori with rotational symmetry. We then review the known examples of embedded minimal surfaces in S^3. Besides the equator and the Clifford t…
We show that for an immersed two-sided minimal surface in R3, there is a lower bound on the index depending on the genus and number of ends. Using this, we show the nonexistence of an embedded minimal surface in R3 of index 2, as conjectured by Choe. Moreover, we show that the index of a immersed two-sided mini…
Finite genus embedded minimal surfaces have limited limit ends.
problem Characterizing limit ends of embedded minimal surfaces.
method Proving properties of limit ends and using them to deduce surface characteristics.
result Embedded minimal surfaces with finite genus have at most two limit ends.
Topology classifies bipolar surfaces; they are not embedded.
problem Classifying bipolar surfaces in the 5-sphere.
method Topological classification and area bounds calculation.
result Bipolar surfaces are not embedded.
Minimal surfaces in a ball have limited area.
problem Bounding the area of genus zero minimal surfaces in a unit ball.
method Proving an area inequality and showing convergence of saturating sequences.
result The area of each nonflat surface is less than its radial projection, with sharp asymptotic bounds.
Classifies minimal surfaces of finite genus in 3D space.
problem Classifying minimal surfaces of finite genus in 3D space.
method Four-step classification, lamination techniques, dynamics theorem, and topology bounds.
result Classification of minimal surfaces of genus zero and infinite topology.
Study of minimal surfaces in 4D with specific ends.
problem Characterize minimal surfaces in R4 with specific ends. method Modification of Costa and Hoffman-Meeks method, generalized Weierstrass representation.
result Minimal surfaces with specific ends are J-holomorphic under certain conditions. Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.
problem Understanding the Morse index of minimal hypersurfaces in real projective spaces.
method Analyzing unstable one-sided and two-sided minimal hypersurfaces in real projective spaces.
result The Morse index of minimal hypersurfaces is at least n+2, with specific examples provided.
We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in S2×R via conjugate constructio…
This paper is the first in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure…
Counterexample disproves conjecture about 3-manifolds.
problem Conjecture about closed Riemannian 3-manifolds without embedded minimal surfaces.
method Provided a counterexample and considered immersed surfaces.
result Conjecture is false for closed surfaces, true for immersed ones.
Unique genus 2 minimal surface in S^3 with specific symmetry group.
problem Finding minimal surfaces in S^3 with specific symmetries.
method Analyzing isometry groups of known minimal surfaces and proving uniqueness.
result ξ_{2,1} is the unique minimal surface of genus 2 with bidihedral symmetry.
We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain natural geometric asymptotic constraints, in terms of certain associated varifold…
We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other ca…