The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
problem Maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
method Established generalized maximum principles and proved stochastic completeness equivalence.
result Stochastic completeness for the heat semigroup is equivalent to generalized maximum principles.
The paper uses graph Laplacians and maximum principles to study learning problems on unknown manifolds.
problem Learning problems on unknown manifolds with noise.
method Maximum principle arguments and techniques from partial differential equations and the Calculus of variations.
result Asymptotic consistency guarantees for noise-corrupted, non-parametric regression.
Study on maximum principles for nonlinear equations on Riemannian manifolds.
problem Investigating strong maximum principles for fully nonlinear equations on Riemannian manifolds.
method Analyzing scaling conditions and applying to various nonlinear operators.
result Established new strong comparison principles for second order uniformly elliptic problems.
We derive, for the square operator of Yau, an analogue of the Omori-Yau maximum principle for the Laplacian. We then apply it to obtain nonexistence results concerning complete spacelike hypersurfaces with constant higher order mean curvature in the Steady State space.
By introducing a weight function to the Laplace operator, Bakry and Émery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the …
Proposes LatLapMED for detecting high-utility anomalies.
problem Detecting statistically rare instances with real-world significance.
method Uses EM algorithm to combine entropy minimization and maximum entropy discrimination.
result Superior performance over existing anomaly detection methods.
We prove that the hypotheses in the version of the Omori-Yau maximum principle that was given by Pigola-Rigoli-Setti are logically equivalent to the assumption that the manifold carries a C2 proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the origin…
Extends strong comparison principle for p-harmonic functions in Carnot-Caratheodory spaces.
problem Proving strong comparison principle for p-harmonic functions in specific geometric settings.
method Extends Bony's propagation of support argument to C^1 solutions of sub-elliptic p-Laplacian.
result Proves strong maximum and comparison principles for p-harmonic functions.
Study strong maximum principles for mean curvature operators on subriemannian manifolds.
problem Investigate strong maximum principles for mean curvature operators on subriemannian manifolds.
method Analyze subriemannian manifolds including Heisenberg groups and cylinders, under Hormander type conditions.
result Show strong maximum principles for horizontal (p-) mean curvature operator and p-(sub)laplacian operator under certain conditions.
The paper establishes gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.
problem Gradient estimates and Liouville theorems for Φ-Laplacian equations on Riemannian manifolds.
method Nonlinear Φ-Bochner formula and Nash-Moser iteration technique for gradient bounds; maximum principle for parabolic case.
result Unified framework for gradient estimates and Liouville theorems for Φ-Laplacian equations.
The paper examines minimal submanifolds with a specific nullity in Euclidean space.
problem Investigating minimal submanifolds with a positive index of relative nullity.
method Analyzing complete Riemannian manifolds and minimal isometric immersions with specific conditions.
result If certain conditions are met, the submanifold must be a cylinder over a minimal surface.
Study potential theory to detect completeness of Finsler manifolds.
problem Detecting completeness of Finsler manifolds via potential theory.
method Potential theoretic aspects of eikonal and infinity Laplace operator, Liouville properties, maximum principles at infinity, viscosity solutions.
result Forward completeness of Finsler manifolds can be detected using Liouville properties and maximum principles at infinity.
A combinatorial version of Yamabe flow is presented based on Euclidean triangulations coming from sphere packings. The evolution of curvature is then derived and shown to satisfy a heat equation. The Laplacian in the heat equation is shown to be a geometric analogue of the Laplacian of Riemannian geometry, although the…
We identify the Variational Principle governing inifinity-Harmonic maps, that is solutions to the Infinity-Laplacian. The system was first derived in the limit of the p-Laplacian as p->inifinity in [K2] and is recently studied in [K3]. Here we show that it is the "Euler-Lagrange PDE" of vector-valued Calculus of Variat…
The paper explores geometric influences on PDE solutions on manifolds.
problem Qualitative behavior of solutions to quasilinear PDEs on Riemannian manifolds.
method Investigates strong and weak maximum principles, compact support principles, and Liouville theorems.
result Identifies thresholds involving curvatures or volume growth to guarantee properties under Keller-Osserman conditions.
Paper establishes maximum principles for weakly 1-coercive operators.
problem Finding conditions for solutions of differential equations to satisfy specific inequalities.
method Maximum principles for weakly 1-coercive operators on Riemannian manifolds.
result Guarantees that solutions of certain differential equations satisfy specific inequalities.
In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and Lp-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scal…
The study examines properties of gradient Yamabe solitons on warped product manifolds.
problem Investigating properties of gradient Yamabe solitons on warped product manifolds.
method Utilizing the maximum principle and modified Li-Yau's technique, the study finds gradient estimates for the warping function.
result The study finds three different gradient estimates for the warping function, one for each sign of the scalar curvature of the fiber manifold.
New characterization of geodesic spheres in space forms.
problem Characterize geodesic spheres in space forms.
method Utilizes the Omori-Yau maximum principle, Walter's formula for mean curvature, and Gårding's inequality.
result Geodesic spheres are the only complete bounded hypersurfaces with constant mean and scalar curvature.
In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some esti…
Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type m…
A new flow-based framework improves graph-based semi-supervised learning while enhancing interpretability.
problem Improving interpretability of semi-supervised learning on graphs.
method Introduces a flow-based learning framework that subsumes and enhances Laplacian-based approaches.
result The flow-based framework improves prediction accuracy without sacrificing interpretability.
Maximizes eigenvalue of sub-Laplacian on CR spheres.
problem Finding the maximum eigenvalue of sub-Laplacian on CR spheres.
method Proving maximum eigenvalue occurs with standard contact form.
result Maximum eigenvalue achieved with standard contact form.
Establishes a boundary maximum principle for varifolds with fixed contact angle.
problem Boundary behavior of varifolds with contact angle constraints.
method Maximum principle for stationary pairs of varifolds with fixed contact angle condition.
result Boundary maximum principle proven for stationary varifolds.
Solves partial data Calderón problem on Riemann surfaces.
problem Calderón problem with partial data on Riemann surfaces.
method Reflection principle applied to connection Laplacian.
result Solves partial data Calderón problem.
Note establishes a local maximum principle for Ricci flow under curvature conditions.
problem Preserving nonnegativity of curvature along Ricci flow with unbounded curvature.
method Combining scaling invariant curvature condition with Dirichlet heat kernel estimates.
result Unified and more direct proof of localized maximum principle.
The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either $\mathds{S}^3$, $\erre\times\mathds{S}^2$ or $\erre^3$, under some very weak conditions on the vector fi…
We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving …
Study proves Maximum Principles for unbounded Riemannian domains.
problem Proving Maximum Principles for unbounded Riemannian domains.
method Examines both ambient manifold and differential operator assumptions.
result Valid Maximum Principles established for unbounded domains.
New Bianchi-convex sets generalize Ricci flow maximum principle.
problem Generalizing maximum principle for Ricci flow.
method Introducing Bianchi-convex sets.
result Hamilton's maximum principle extended to Bianchi-convex sets.
In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…
Study extends ODE Maximum Principle to non-compact hypersurfaces in hyperbolic space.
problem Analyzing long-term behavior of IMCF on non-compact hypersurfaces.
method Extends ODE Maximum Principle to non-compact hypersurfaces using Omari-Yau maximum principle at infinity.
result Showed long-time existence and asymptotic convergence of IMCF to horospheres.
A principle for minimal surfaces in any dimension and codimension.
problem Boundary maximum principle for minimal submanifolds.
method Established a boundary maximum principle for minimal submanifolds in arbitrary codimension.
result Generalized boundary maximum principle to any dimension and codimension.
Proves a principle for one-phase Bernoulli problem minimizers.
problem One-phase Bernoulli problem minimizers.
method Strong maximum principle, Alt-Caffarelli functional, Hardt-Simon-type foliation.
result Constructs a foliation for global minimizers.
Derives a parabolic maximum principle for mean curvature flow.
problem Maximum principle for mean curvature flow in curved spaces.
method Derives a parabolic Omori-Yau maximum principle.
result Maximum principle preserves image of Gauss map and self-shrinkers.
The paper generalizes a curvature result for hyperbolic affine hyperspheres.
problem Curvature properties of hyperbolic affine hyperspheres.
method Tensorial maximum principle applied to the Bakry-Émery tensor.
result Metric measure spaces have non-positive Bakry-Émery tensor.
In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral …
In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the form Lu≥b(x)f(u)ℓ(∣∇u∣) and Lu≥b(x)f(u)ℓ(∣∇u∣)−g(u)h(∣∇u∣), where L is a non-linear diffusion-type operator. Prototypical ex…
The paper extends a Maximum Principle for hypersurfaces in R^(n+1) with bounded mean curvature.
problem Generalizing a Maximum Principle for hypersurfaces in R^(n+1) with bounded mean curvature.
method Extending a Maximum Principle at Infinity for disjoints hypersurfaces in R^(n+1) with bounded mean curvature.
result The extension of the Maximum Principle for hypersurfaces in R^(n+1) with bounded mean curvature.
We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is comp…
In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity v…
This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operato…
The paper proves gap results for self-shrinkers in r-mean curvature flow.
problem Understanding the gap in properties of self-shrinkers in r-mean curvature flow. method Proving gap results using a modified second fundamental form and a differential operator.
result Proper self-shrinkers are parabolic for a certain second-order differential operator.
This note introduces a duality principle for nonlinear equations.
problem Maximum principles at infinity for nonlinear equations.
method Ahlfors property and Khas'minskii potentials.
result Unified framework for various maximum principles.
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
Study compares nodal sets of solutions to the Allen-Cahn equation.
problem Comparing nodal sets of solutions to the Allen-Cahn equation with conical asymptotics.
method Maximum principle for linearized operator on unbounded domains.
result Positive phase uniquely determines the solution and enforces global ordering.
Novel framework for graph learning from data under structural and Laplacian constraints.
problem Graph learning from data under structural and Laplacian constraints.
method Formulation of graph learning problems, probabilistic interpretations, and specialized algorithms incorporating graph Laplacian and structural constraints.
result Experimental results show the proposed algorithms outperform state-of-the-art methods.