Proves minimal crossing diagrams for specific spatial graphs.
problem Proving minimal crossing diagrams for spatial graphs.
method Analyzing adequate diagrams and replacing vertices and edges.
result All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.
The paper proves removable singularity for nonlocal minimal graphs.
problem Proving removable singularities for nonlocal minimal graphs.
method Analyzing (s,1)-capacity zero compact sets to ensure graphs are minimal in the entire domain. result Nonlocal minimal graphs are removable in the entire domain if they are minimal in a set of (s,1)-capacity zero. Minimal graphs over simply connected domains grow at most exponentially.
problem Growth of minimal graphs over simply connected domains with boundary values 0.
method Analyzing solutions to the minimal surface equation.
result Minimal graphs have at most exponential growth.
Develops a method to construct entire minimal graphs of odd dimensions.
problem Constructing entire minimal graphs of odd dimensions and arbitrary codimensions.
method Evolving-plane ansatz reducing minimal surface system to geodesic equation on Grassmannian.
result Yields a rich family of explicit entire minimal graphs of odd dimension and arbitrary codimension.
Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
The paper studies harmonic graphs in the Heisenberg group and their properties.
problem No analogous theorem exists for H-minimal surfaces in the Heisenberg group. method Introduced intrinsic Dirichlet energy and studied its critical points (contact harmonic graphs).
result Calibration condition and construction of energy-minimizing graphs with various singularities.
Minimal graph level sets are concave if boundary is concave.
problem Understanding curvature of minimal graph level sets.
method Proved an inequality and showed geometric properties.
result Level sets of minimal graphs are concave if boundary is concave.
Sharp upper bound for minimal graph area in unit ball established.
problem Determining the exact upper limit for the area of minimal graphs intersecting a unit ball.
method Constructing a sequence of minimal graphs via solutions to a Dirichlet problem.
result The areas of constructed minimal graphs tend to the upper bound of 2π. We extend Osserman's lemma on the generalized Gauss map of two-dimensional minimal graphs of higher codimension, construct a Jenkins-Serrin type special Lagrangian Scherk graph explicitly, and generalize Calabi's correspondence between minimal graphs and maximal graphs.
Two new minor minimal intrinsically chiral graphs identified.
problem Identifying intrinsically chiral graphs in molecular structures.
method Analyzing graph symmetry and embedding properties.
result Found two new minor minimal intrinsically chiral graphs Γ7 and Γ8. We prove that a strictly stable minimal Ch2 intrinsic graph G is locally area-minimizing, i.e. given any Ch1 graph S with the same boundary, Area(G)<Area(S) unless G=S. As a consequence we show the existence and the uniqueness of C∞ minimal graphs with prescribed small boundary datum…
Paper estimates Gaussian curvature of minimal graphs in a specific manifold.
problem Estimating Gaussian curvature of minimal graphs in MimesR. method Using Weierstrass representation via ℘−harmonic mappings and Schwarz lemma type results. result Proves Schwarz lemma type and Heinz type results for harmonic mappings.
Paper proves flatness of anisotropic minimal graphs in half-spaces.
problem Anisotropic minimal graphs with free boundaries in half-spaces.
method Proves flatness using linear growth conditions.
result Anisotropic minimal graphs in half-spaces are flat if they have at most one-sided linear growth.
New forms calibrate minimal graphs in arbitrary dimensions.
problem Calibrating minimal graphs in arbitrary codimension.
method Constructing closed forms from minimal graphs and estimating their comass.
result Conditions ensuring minimal graphs are calibrated and area-minimizing.
Paper solves long-standing Gaussian curvature conjecture for minimal graphs.
problem Gaussian curvature of minimal graphs over the unit disk.
method Complex-analytic methods, conformal harmonic parameterization.
result Sharp estimate for Gaussian curvature at the origin of minimal graphs.
Anisotropic minimal graphs over half-spaces are flat.
problem Characterizing minimal graphs over half-spaces.
method Maximum principle and fully nonlinear PDE theory.
result Anisotropic minimal graphs over half-spaces are flat.
We find the minimal number of links in an embedding of any complete k-partite graph on 7 vertices (including K7, which has at least 21 links). We give either exact values or upper and lower bounds for the minimal number of links for all complete k-partite graphs on 8 vertices. We also look at larger complete bip…
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
We prove that a family of entire intrinsic minimal graphs in the Heisenberg group are not perimeter minimizing.
We prove surfaces are unknotted with specific properties.
problem Unknottedness of free boundary minimal surfaces and self-shrinkers.
method Introduced concepts of boundary graph and graph at infinity to prove unknottedness.
result Proved surfaces are unknotted with specific properties.
New bound for group action length without diameter restriction.
problem Bounding minimal translation length for Artin groups.
method Graph theoretic properties of biconnected graphs.
result Upper bound of 2 for minimal translation length holds without diameter restriction.
It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minim…
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible). There is a natural necessary condition on such grap…
Bernstein theorem proven for 2-valued minimal graphs in 4D.
problem Classifying 2-valued minimal graphs in 4D.
method Analyzing blowdown cones and combinatorial arguments.
result Two-valued minimal graphs in 4D are unions of two 3D planes.
In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be g…
Paper classifies minimal graph transformations into new families of surfaces.
problem Classifying minimal graph transformations into new families of surfaces.
method Formulated and solved a coupled system of partial differential equations, reduced to solving an ordinary differential equation.
result Established rigorous equivalence to a modified problem for a harmonic function, yielding new families of minimal surfaces.
Minimal graph theorem proven for convex domains.
problem Characterizing minimal graphs over convex domains.
method Analyzing minimal surface equation solutions on convex domains.
result Minimal graphs over convex domains are linear.
We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links for K4,4,1…
We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil_3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil_3 is entire. Also, the si…
Extends Smale's principle to produce minimal graphs with singularities.
problem Creating minimal graphs with isolated singularities in higher dimensions.
method Extends Smale's singular bridge principle to arbitrary codimension and applies it to specific minimal cones.
result Produces a minimal graph in 7D with any number of isolated singularities.
Sharp Liouville theorem for minimal graphs on manifolds with nonnegative Ricci curvature.
problem Characterizing smooth solutions to minimal hypersurface equations on manifolds with nonnegative Ricci curvature.
method Gradient estimate for minimal graphs over Σ with small linear growth of the negative parts of graphic functions via iteration. result Every smooth solution u to minimal hypersurface equation on Σ is a constant provided u has sublinear growth for its negative part. We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper "A martingale approach to minimal surfaces" to show that Brownian motion on such a minimal …
Study minimal graphs on non-negative Ricci curvature manifolds.
problem Minimal graphs with linear growth on manifolds with non-negative Ricci curvature.
method New gradient estimate for minimal graphs and heat equation techniques.
result Non-constant minimal graphs force tangent cones to split off a line.
Unlike R3, the homogeneous spaces E(−1,τ) have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantees that a minimal surface in E(−1,τ) is such a graph. More specifically: we introduce the definition of a generalized slab in $\mathbb{E…
The paper proves stability of certain graph types in Euclidean space with specific densities.
problem Stability of vertical and radial graphs in Euclidean space with certain densities.
method Techniques of calibrations used to prove stability and minimization.
result Vertical and radial graphs are strongly stable for specific densities.
We consider adaptations of the Mumford-Shah functional to graphs. These are based on discretizations of nonlocal approximations to the Mumford-Shah functional. Motivated by applications in machine learning we study the random geometric graphs associated to random samples of a measure. We establish the conditions on the…
We study a problem of geometric graph theory: We determine the triply periodic graph in Euclidean 3-space which minimizes length among all graphs spanning a fundamental domain of 3-space with the same volume. The minimizer is the so-called srs network with quotient the complete graph on four vertices K4. The network…
This paper classifies chiral graphs up to size 12.
problem Understanding the chirality of simple graphs to predict molecular behavior.
method Classifying minor minimal intrinsically chiral graphs among simple graphs of size up to 12.
result Complete set of minor minimal graphs for intrinsic properties of chiral molecules.
We show that the 20 graph Heawood family, obtained by a combination of triangle-Y and Y-triangle moves on K7, is precisely the set of graphs of at most 21 edges that are minor minimal for the property not 2--apex. As a corollary, this gives a new proof that the 14 graphs obtained by triangle-Y moves on K7 are t…
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…
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.
Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.
problem Finding extremal minimal graphs with zero Gaussian curvature at the center.
method Analyzing Scherk surfaces and their properties.
result Scherk type minimal surfaces are extremals for zero-curvature minimal graphs.
Minimal graphs grow slowly on curved spaces, proving constant solutions.
problem Characterizing minimal graphs with sublinear growth on manifolds.
method New technique to get gradient bounds by integral estimates, no further geometric assumptions.
result Entire solutions are constant when negative part grows like r/logr. Examples of area-minimizing graphs with low regularity in a specific group.
problem Finding area-minimizing graphs with low regularity in a sub-Finsler Heisenberg group.
method Providing examples of entire area-minimizing horizontal graphs with prescribed singular sets.
result Examples of area-minimizing graphs that are locally Lipschitz but not necessarily smoother.
We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence…
GALA framework learns invariant graph representations via environment augmentation with minimal assumptions.
problem Learning invariant graph representations from different environments without additional assumptions.
method Developed GALA framework with minimal assumptions of variation sufficiency and consistency. Uses an assistant model to differentiate graph environment changes.
result Extracting maximally invariant subgraphs to proxy predictions identifies underlying invariant subgraphs for successful out-of-distribution generalization.
We provide a characterization for complex analytic curves among two-dimensional minimal graphs in R4 via the Jacobian
New IPL graphs identified and conditions for their projective embeddings established.
problem Characterizing and identifying intrinsically projectively linked graphs.
method Applying Δ-Y exchanges and analyzing projective planar graphs.
result No minor-minimal IPL graphs on 16 edges exist, and new ones are identified.