This study proves the Half Space Property for RCD(0,N) and RCD(K,N) spaces.
problem Proving the Half Space Property for RCD(K,N) spaces.
method Analyzing locally perimeter minimizing sets and extending Green's functions results.
result The Half Space Property holds for RCD(K,N) spaces under specific conditions.
Proves intersection properties of minimal hypersurfaces in various spaces.
problem Intersection properties of minimal hypersurfaces in different geometric settings.
method Two approaches: classifications of stable minimal hypersurfaces and conformal change with comparison geometry.
result Intersection properties for minimal hypersurfaces in specific geometric settings, including free boundary minimal hypersurfaces.
The study proves properties of capillary graphs in half-spaces.
problem Characterizing capillary minimal graphs in half-spaces.
method Analyzing tangent cones and regular set properties.
result Capillary minimal graphs in low dimensions or specific cone conditions are linear.
Paper studies minimal surfaces in curved spaces, proving existence and properties.
problem Existence and properties of minimal surfaces with free boundaries.
method Degree theory, Banach manifold construction, Fredholm map analysis.
result Space of partially free boundary minimal half disks is a Banach manifold.
Concrete distribution properties examined on simplex.
problem Properties of Concrete distribution on simplex.
method Reflection and location-scale transformation of uniform distribution; explicit parameterization to Poincaré half-space.
result Fisher information and information metric are hyperbolic space; Fisher-Rao geodesic distance computed.
Study on Banach half-Lie groups and their properties.
problem Characterizing and understanding the properties of half-Lie groups in infinite dimensions.
method Investigation of Banach half-Lie groups, their extensions, and right invariant strong Riemannian metrics.
result The full Hopf--Rinow theorem holds for Banach half-Lie groups, a surprising result.
In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci…
The paper defines capacities for minimal graphs over manifolds and proves the half-space property.
problem Characterizing minimal graphs and their properties over manifolds.
method Defining capacities using relative volume, studying solutions of bounded variation, and analyzing boundary behavior.
result Proves the half-space property for M-parabolic manifolds. Ancient solutions found for a specific flow on symplectic half-flat structures.
problem Existence of solutions for a particular geometric flow.
method Analyzes Type IIA flow on symplectic half-flat SU(3)-structures.
result Existence of ancient, immortal, and eternal solutions under suitable conditions.
Study on umbilical submanifolds in specific geometric spaces.
problem Characterizing curvature properties of umbilical submanifolds.
method Analyzing totally and screen totally umbilical half lightlike submanifolds in almost contact B-metric manifolds.
result Curvature properties of these submanifolds were studied and characterized.
The paper explores properties of 1-surfaces in hyperbolic space and proves strong half-space theorems.
problem Investigating properties of 1-surfaces in Riemannian manifolds with specific curvature bounds.
method Analyzes the intersection problem for 1-surfaces with bounded curvature in a complete Riemannian three-manifold with Ricci curvature bounded from below.
result Strong half-space theorems for complete 1-surfaces in hyperbolic space with bounded curvature.
Recently dictionary screening has been proposed as an effective way to improve the computational efficiency of solving the lasso problem, which is one of the most commonly used method for learning sparse representations. To address today's ever increasing large dataset, effective screening relies on a tight region boun…
Study variational properties of curves in half-plane with area constraints.
problem Characterize critical points of inverse mean curvature.
method Variational analysis of curves with boundary constraints.
result Existence and stability of critical points with prescribed area.
In this paper, we prove a half-space theorem with respect to constant mean curvature 1/2 entire graphs in E(−1,τ). If Σ is such an entire graph and Σ′ is a properly immersed constant mean curvature 1/2 surface included in the mean convex side of Σ then Σ′ is a vertical translate of Σ. We also h…
It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Loja…
The paper establishes inequalities for p-capacitary functions in flat half-spaces.
problem Understanding p-capacitary functions in asymptotically flat half-spaces. method Establishes monotone quantities and mass-capacity inequalities.
result Sharp inequalities attain equality on a Schwarzschild half-space.
Developed a half-space model for pseudo-hyperbolic space.
problem Modeling pseudo-hyperbolic space for any dimensions.
method Created an isometric embedding of pseudo-hyperbolic space into a half-space.
result Geodesics, totally geodesic submanifolds, horospheres, and isometry group are described in the half-space model.
Paper classifies critical points in half-space with new distance function.
problem Classifying critical points in half-space with capillary CMC hypersurfaces.
method New shifted distance function for capillary problem in half-space.
result Proves Alexandrov-type theorem for singular capillary CMC hypersurfaces.
The paper characterizes Ricci solitons on the Poincaré upper half plane.
problem Characterizing Ricci solitons on the Poincaré upper half plane.
method Classifying and generalizing Ricci solitons and soliton equations in the half plane of Poincaré.
result Obtained some properties of solitons about their geodesic flows.
Paper studies stability of curved surfaces in a half-space.
problem Stability of anisotropic capillary hypersurfaces in a half-space.
method Analyzes weak stability and proves Bernstein-type theorems.
result Compact hypersurfaces are stable if and only if they are a truncated Wulff shape.
Study metrics on half plane with specific curvature properties.
problem Warped product metrics on half plane.
method Holomorphic isometries and sectional curvature analysis.
result Metrics with zero and unbounded negative curvature exist.
We construct higher genus Riemann's minimal surfaces properly embedded in the Euclidean space. To do that we glue end by end a Costa-Hoffman-Meeks examples to two halves genus zero Riemann's minimal surfaces. In first we need to perform a deformation of a Costa-Hoffman-Meeks example to prescribe the flux vector along t…
Paper introduces capillary Schwarz symmetrization in half-space.
problem Capillary problems in half-space.
method Introduces a special anisotropic gauge to transform capillary symmetrization to convex symmetrization.
result Capillary Schwarz symmetrization in half-space established.
We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable f-harmonic function with controlled gradient. In the immersed case, a new direct proof …
In this work we study the geodesic structure of the space Σ(X) of compact balls of a complete and locally compact metric length space endowed with the Hausdorff distance dH. In particular, we focus on a geometric condition (referred to as the shooting property) that enables us to give an explicit isometry between …
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.
Study proves no minimal surfaces can be contained in certain half-spaces or cones.
problem Prohibiting minimal surfaces from certain geometric configurations.
method Analyzes weighted minimal surfaces in R3 with height-dependent weights. result No proper surfaces can be contained in specific half-spaces or cones.
In this paper we study the maximal stable domains on minimal catenoids in Euclidean and hyperbolic spaces and in H2×R. We in particular investigate whether half-vertical catenoids are maximal stable domains (\emph{Lindelöf's property}). We also consider stable domains on catenoid-cousins in hyperbolic space. …
We propose a simple proof of the vertical half-space theorem for Heisenberg space.
Study on minimal surfaces with free boundary in a half-space, improving index estimates.
problem Non-existence of index two embedded minimal surfaces with free boundary in a half-space.
method Improved estimates of Neumann and Dirichlet indices, simplified proof of lower bounds.
result Answered Ambrozio et al.'s question and provided new lower bounds.
This note proves a Gaussian version of a Pólya-Szegö conjecture using rearrangement techniques.
problem Finding the domain with the minimum Gaussian principal frequency when the Gaussian torsional rigidity is fixed.
method Adapted Kohler-Jobin rearrangement technique to the Gauss space, considering a modified torsional rigidity and rearranging layers to half-spaces.
result The Gaussian principal frequency is minimized for the half-space when the Gaussian torsional rigidity is fixed.
Study on 4D Ricci solitons with specific curvature properties.
problem Characterizing 4D complete gradient shrinking Ricci solitons with half positive isotropic curvature.
method Curvature estimates, strong maximum principle, classification arguments.
result New classification results for gradient shrinking Kähler-Ricci solitons and 4D complete gradient shrinking Ricci solitons with half nonnegative isotropic curvature.
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.
In this paper, we analyze the Bollobás and Riordan polynomial R for ribbon graphs with half-ribbons introduced in [Combinatorics, Probability and Computing 31, 507-549, 2022]. We prove the universality property of a multivariate version of R whereas R itself turns out to be universal…
Paper solves inequalities for capillary hypersurfaces in half-spaces.
problem Finding inequalities for convex capillary hypersurfaces in half-spaces.
method Introduced quermassintegrals and constructed a new locally constrained curvature flow to prove convergence to spherical caps.
result Obtained Alexandrov-Fenchel inequalities for convex capillary hypersurfaces.
New theorem for nonlocal minimal surfaces in any dimension.
problem Classical half-space theorems for minimal surfaces.
method Established a new half-space theorem.
result Result holds in any dimension.
We investigate the SL(2,R) invariant geodesic curves with the as- sociated invariant distance function in parabolic geometry. Parabolic geom- etry naturally occurs in the study of SL(2,R) and is placed in between the elliptic and the hyperbolic (also known as the Lobachevsky half-plane and 2- dimensional Minkowski half…
New sub-Riemannian spaces with boundary meet curvature-dimension condition.
problem Finding sub-Riemannian manifolds with boundary satisfying curvature-dimension condition.
method Constructing specific sub-Riemannian structures on half-spaces and hemispheres.
result Provided new examples of sub-Riemannian manifolds with boundary that meet RCD(K,N) condition. Optimal Liouville theorem for half-Euclidean space equations.
problem Optimal Liouville-type theorems for conformally invariant equations.
method Established optimal Liouville-type theorems for conformally invariant second-order elliptic equations.
result Proved an optimal Liouville-type theorem for equations in the half-Euclidean space.
New Minkowski inequality for capillary surfaces in half-space.
problem Establishing a new Minkowski inequality for capillary surfaces.
method Flow of inverse mean curvature type for capillary hypersurfaces in a half-space.
result Derive a new Minkowski-type inequality for star-shaped and mean convex capillary hypersurfaces.
Study on capillarity minimizers with nonlocal repulsion and gravity, proving existence and nonexistence.
problem Minimizing an energy functional with capillarity, nonlocal repulsion, and gravity.
method Quantitative isoperimetric inequalities applied to capillarity problem in a half-space.
result Existence and nonexistence of minimizers for various nonlocal kernels and masses.
Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.
problem Extending Wente's result to anisotropic capillary surfaces.
method New Heintze-Karcher inequality and Minkowski formula.
result Anisotropic capillary hypersurfaces in half-spaces are Wulff shapes.
We study second-order PDEs in 4D for which the conformal structure defined by the characteristic variety of the equation is half-flat (self-dual or anti-self-dual) on every solution. We prove that this requirement implies the Monge-Ampere property. Since half-flatness of the conformal structure is equivalent to the exi…
Alternative proof for ribbon surfaces in 3D space.
problem Proving ribbonness of surfaces in 3D space.
method Using a compact oriented proper surface in upper half 4-space.
result Link bounds a ribbon surface in upper half 4-space.
Inverse metric matrices on Siegel-Jacobi spaces are calculated for Berezin quantization.
problem Calculating inverse metric matrices on Siegel-Jacobi spaces.
method Inversion of metric matrices on XnJ and ildeXnJ. result Explicit calculations of inverse metric matrices for n=2. Proposes NUV priors for half-space and box constraints.
problem Adding constraints to linear Gaussian models without computational cost.
method Introduces NUV representations for half-space and box constraints.
result Adds constraints to linear Gaussian models without affecting computational tractability.
Two minimal hypersurfaces in a ball intersect in any half-ball.
problem Intersection properties of minimal hypersurfaces in a ball.
method Analyzing the intersection of two minimal hypersurfaces in a unit Euclidean ball.
result Intersection point in any half-ball, strong Frankel property.
Paper analyzes iterative learning for concept classes and learns half-spaces.
problem Learning concept classes efficiently with iterative learners.
method Analyzes various settings of iterative learning and provides a constructive algorithm for half-spaces.
result Constructive iterative algorithm for learning half-spaces from informant.