Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.
problem Estimating regression functions on unknown smooth submanifolds.
method Random geometric graph structure, Bayesian priors based on random basis expansion in graph Laplacian eigenbasis.
result Posterior contraction rates are minimax optimal for any positive smoothness index.
Study area and coarea formulas for graphs and submanifolds in Carnot groups.
problem Understanding geometric properties of submanifolds in Carnot groups.
method Developed area and coarea formulas for CH1 intrinsic graphs and submanifolds. result Deduced density properties for Hausdorff measures and coarea formula for Carnot groups.
The paper proves properties for random graphs based on geometric submanifolds.
problem Establishing measure-metric properties of random geometric graphs.
method Analyzing ε-neighborhood graphs with specific conditions on submanifold and distribution. result Volume doubling and local Poincaré inequalities hold for random geometric graphs with high probability.
New hyperbolic graph constructed from projections of free splitting graph.
problem Constructing a new hyperbolic graph from projections of free splitting graph.
method Using submanifold projections and geometric realization of free splitting graph.
result A new hyperbolic graph constructed for n≥3. Graph Laplacian converges to Laplace-Beltrami operator with a specific rate.
problem Convergence of graph Laplacian to Laplace-Beltrami operator on random geometric graphs.
method Analysis of random geometric graphs and eigenvalue convergence rates.
result Eigenvalues and eigenvectors of graph Laplacian converge to Laplace-Beltrami operator with rate O((nlogn)2m1). New method uses manifold learning to infer latent positions of 1D submanifolds in random dot product graphs.
problem Inference on latent positions of unknown 1D submanifolds in RDPGs.
method Apply Isomap for manifold learning to estimate arc lengths on the unknown submanifold.
result Test statistics based on Isomap converge to known submanifold power as auxiliary vertices increase.
Graph Laplacian spectral convergence rate established for manifold learning.
problem Consistency of spectral clustering algorithms on manifold data.
method Established spectral convergence rate of graph Laplacian for a random sample of a d-dimensional compact submanifold.
result Consistency of spectral clustering algorithms via spectral convergence rate.
Model for associative submanifolds in K3 fibrations.
problem Understanding singularity formation in associative submanifolds.
method Graphs in a 3-manifold with locally gradient flow lines.
result Produces analogues of known singularity formation phenomena.
This is mainly a survey article on the recent development of the theory of graph-like Legendrian unfoldings and its applications. The notion of big Legendrian submanifolds was introduced by Zakalyukin for describing the wave front propagations. Graph-like Legendrian unfoldings belong to a special class of big Legendria…
Study solves Dirichlet problem for higher-dimensional submanifolds.
problem Existence and uniqueness of graphical maximal submanifolds.
method General existence and uniqueness results for any codimension.
result General existence and uniqueness results for graphical maximal submanifolds of higher codimension.
We show the volume maximizing property of the special Lagrangian submanifolds of a pseudo-Euclidean space. These special Lagrangian submanifolds arise locally as gradient graphs of solutions to Monge-Ampere Equations.
This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.
Signals are submanifolds; bounds on energy calculated.
problem Abstract theory of signal propagation.
method Energy inequalities and bounds calculated for specific signal spaces.
result Upper and lower bounds on energy derived for various signal configurations.
Paper analyzes convergence of Laplacian eigenmaps on submanifolds with singularities.
problem Analyzing convergence of Laplacian eigenmaps on submanifolds with singularities.
method Using ε-neighborhood graphs constructed from random points on the submanifold, the paper provides a spectral approximation result for the Laplacian.
result The convergence rate for the eigenvalue of the Laplacian is \( O\left(\left(\log n/n
ight)^{1/(m+2)}
ight) \), where \( m \) and \( n \) are the dimension of the manifold and the sample size, respectively.
A new model clusters networks with community-specific submanifold structures.
problem Clustering networks with community-specific submanifold structures.
method Latent Structure Block Models (LSBM) for Bayesian spectral graph clustering.
result LSBM correctly recovers underlying communities in one-dimensional manifold structures.
The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.
problem Bernstein problem for smooth maps to lower dimensions forming calibrated submanifolds.
method Established conditions for maps to be affine based on the slope's second elementary symmetric polynomial.
result Conditions ensuring maps are affine for coassociative and Cayley submanifolds in R7 and R8. We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.
The paper studies the asymptotic expansion of Gaussian integral operators on Riemannian submanifolds.
problem Analyzing the asymptotic behavior of Gaussian integral operators on Riemannian submanifolds.
method Deriving a full asymptotic expansion of the Gaussian integral operator and computing the first-order correction term.
result Explicit computation of the first-order correction term in terms of mean curvature vector and scalar curvature.
Study of Lagrangian submanifolds in para-complex Euclidean space.
problem Characterizing and understanding Lagrangian submanifolds in para-complex Euclidean space.
method Analyzing curvature equations and extrinsic geometry of submanifolds.
result Characterization of minimal Lagrangian surfaces and self-similar solutions of Mean Curvature Flow.
Special Lagrangian submanifolds emerge from K3 surface collapse.
problem Understanding special Lagrangian submanifolds in K3 surface collapse.
method Lifting affine lines to degenerating sequences of special Lagrangian submanifolds.
result Constructing special Lagrangian two-spheres connecting Taub-NUT bubbles.
A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms …
Paper proves stability of flow in cotangent bundle for special Lagrangian submanifolds.
problem Stability of generalized Lagrangian mean curvature flow in cotangent bundle.
method New derivative estimates to weaken initial conditions and remove curvature constraints.
result Stability of flow near special Lagrangian submanifolds in cotangent bundle.
Proves smoothness of certain Lagrangian submanifolds in complex space.
problem Smoothness of Hamiltonian stationary Lagrangian submanifolds in complex space.
method Morrey-type theorem and analysis of nonlinear fourth order equation.
result Proves smoothness of weakly harmonic Lagrangian phase submanifolds.
An n dimensional minimal submanifold Σ of Rn+m is called non-parametric if Σ can be represented as the graph of a vector-valued function f:D⊂Rn↦Rm. This note provides a sufficient condition for the stability of such Σ in terms of the norm of the differential df.
Study on mean curvature flow of graphs in higher dimensions.
problem Analyzing the evolution of graphs under mean curvature flow.
method Derives estimates using a new maximum principle for submanifolds, applies to uniformly area decreasing maps.
result Graphicality and area decreasing property are preserved for uniformly area decreasing maps.
The paper extends a Bernstein theorem to codimension 2 minimal submanifolds.
problem Proving a Bernstein theorem for minimal submanifolds in higher codimension.
method Using convexity properties of Grassmannians and Allard's theorem.
result Minimal submanifolds in codimension 2 must be planes.
We construct examples of C∞ smooth submanifolds in Cn and Rn of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianaly…
Let Ωand \tildeΩ be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: Ω\to \tildeΩ such that the graph Σ= \{(x,f(x)): x \in Ω\} is a minimal Lagrangian submanifold of \mathbb{R}^n \times \mathbb{R}^n.
A standard fact about two incompressible surfaces in an irreducible 3-manifold is that one can move one of them by isotopy so that their intersection becomes π1-injective. By extending it on the maps of some 3-dimensional Zn-manifolds into 4-manifolds, we prove that any homotopy equivalence of 4-dimensio…
We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.
Constructs special Lagrangian submanifolds in Calabi-Yau 3-folds.
problem Constructing special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds.
method Constructs special Lagrangian submanifolds in collapsing Calabi-Yau 3-folds fibered by K3 surfaces.
result Special Lagrangian submanifolds shrink to 1-dimensional graphs in the base as the 3-folds collapse.
Geodesic connects Lagrangian graphs over torus in complex space.
problem Existence of geodesic connecting Lagrangian graphs in complex space.
method Formulated as a degenerate elliptic equation, solved via Dirichlet problem.
result Geodesic connecting Lagrangian graphs can be constructed.
The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.
problem Calculating area Siegel--Veech constants for affine invariant submanifolds of REL zero.
method Using volumes of the principal boundary strata and intersection theory.
result Proves a conjectural formula for the area Siegel--Veech constant in the case of REL zero.
The Gromov-Eliashberg theorem says that the group of symplectomorphisms of a symplectic manifold is C^0-closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It is also known that such Lagrangian submanifolds are locally…
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.
Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore…
Proves curvature estimates for 3D surfaces solving scalar curvature equations.
problem Estimating curvature of 3D surfaces solving scalar curvature equations.
method Integral method and new Lagrangian submanifold observation.
result Interior curvature estimates for 3D hypersurfaces are proven.
Minimal surfaces with complex branching structures constructed using various methods.
problem Construct minimal surfaces with specific branching sets.
method Three constructions using perturbation, barrier methods, and bifurcation theory.
result Minimal surfaces with stratified branching sets constructed as graphs of two-valued functions.
Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is the amalgamation of strongly irreducible Heegaard splittings. The splitting surfaces S_i of these strongly irreducible Heegaard splittings have t…
The paper extends graph embedding models to handle multiple relations.
problem Link prediction in multi-relational networks.
method Generalized pseudo-Riemannian embedding models to multi-relational networks, considering relations as submanifolds.
result Validation of the approach in link prediction tasks, including knowledge graph completion and biological domain analysis.
This is a survey of our work on spacelike graphic submanifolds in pseudo-Riemannian products, namely on Heinz-Chern and Bernstein-Calabi results and on the mean curvature flow, with applications to the homotopy of maps between Riemannian manifolds.
Study examines mean curvature flow on graphs of maps between manifolds.
problem Investigating mean curvature flow on graphs of maps between manifolds with bounded geometry.
method Investigates the mean curvature flow of graphs of smooth length-decreasing maps f:RmoN. result Uniform decay estimates for all derivatives of order ≥2 of ft along the flow. Maps between 2D spaces evolve under area-decreasing conditions.
problem Understanding the evolution of maps under area-decreasing constraints.
method Mean curvature flow of the graph of a map between 2D Euclidean spaces.
result Existence and uniform decay estimates for the evolving submanifold.
Let (Σ, ω) be a compact Riemann surface with constant curvature c. In this work, we proved that the mean curvature flow of a given Hamiltonian diffeomorphism on Σ provides a smooth path in Ham(Σ), the group of all Hamiltonian diffeomorphisms of Σ. This result gives a proof, in the case of graph of Hamiltonian diffeomor…
We extend the work of Simon and Wickramasekera, who constructed a large class of C1,μ multivalued solutions to the minimal surface equation, to produce C1,μ multivalued solutions to more general classes of elliptic equations and systems, including the minimal surface system with small boundary data and the La…
Let N be a complete, simply-connected surface of constant curvature κ\leq 0. Moreover, suppose that Ωand \tildeΩ are strictly convex domains in N with the same area. We show that there exists an area-preserving diffeomorphism from Ωto \tildeΩ whose graph is a minimal submanifold of N \times N.
Study Hodge Laplacians for manifold data, improving error bounds.
problem Approximating Laplace-Beltrami operator on differential forms.
method Higher-order graph Laplacians (Hodge Laplacians) as approximations.
result High-probability error bound for Dirichlet forms.
Suppose that C=(C1,...,Cm) is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold (X,ω) with connected, negative definite intersection graph ΓC. We show that by replacing an appropriate neighborhood of ∪Ci with a smoothing WS of a normal surface singularity (S,0) w…