Study classifies semi-discrete linear Weingarten surfaces with Weierstrass-type representations and analyzes their singularities.
problem Characterizing semi-discrete linear Weingarten surfaces with Weierstrass-type representations and their singularities.
method Established properties, classified, and analyzed the singularities of semi-discrete linear Weingarten surfaces in Riemannian and Lorentzian spaceforms.
result Defined and analyzed the singularities of semi-discrete linear Weingarten surfaces, including those with non-zero constant Gaussian curvature, parallel surfaces of minimal and maximal surfaces, and constant mean curvature 1 surfaces in de Sitter 3-space.
Study of hypersurfaces in curved spaces with specific curvature properties.
problem Characterizing hypersurfaces in spaces of constant curvature with particular curvature properties.
method Investigates hypersurfaces isometrically immersed in semi-Riemannian spaces of constant curvature, focusing on the curvature tensor and its properties.
result Hypersurfaces in the specified spaces satisfy a Roter type equation, linking their curvature tensor to specific tensor products.
In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work \cite{Mon, Brendle,BE} considering constant mean curvature functions. Secondly, …
In this work, we study spacelike surfaces in Minkowski space E13 foliated by pieces of circles and that satisfy a linear Weingarten condition of type aH+bK=c, where a,b and c are constant and H and K denote the mean curvature and the Gauss curvature respectively. We show that such surfaces must be surfa…
We give a relatively simple proof that a translation surface in Euclidean space that satisfies a relation of type aH+bK=c, for some real numbers a,b,c, where H and K are the mean curvature and the Gauss curvature of the surface, respectively, must have a=0 or b=0, and thus, K is constant or H is constan…
New example shows multisymplectic forms without Darboux charts.
problem Existence of multisymplectic forms without Darboux charts.
method Construction of multisymplectic three-forms on R6. result Provided an explicit counterexample to Darboux theorem in multisymplectic framework.
A twisted curve in Euclidean 3-space E^3 can be considered as a curve whose position vector can be written as linear combination of its Frenet vectors. In the present study we study the twisted curves of constant ratio in E^3 and characterize such curves in terms of their curvature functions. Further, we obtain some re…
We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respect…
Study bounds CMC surface index in 3-manifolds using energy.
problem Bounding the index of CMC surfaces in 3-manifolds.
method Energy comparison to prove linear upper bound.
result Linear upper bound on CMC surface index.
The paper proves rigidity for hypersurfaces with constant shifted curvature functions in warped product manifolds.
problem Characterizing and proving rigidity for hypersurfaces with constant shifted curvature functions.
method Using integral inequalities and Minkowski-type formulas, the paper derives rigidity theorems in sub-static warped product manifolds.
result The paper provides new characterizations and rigidity results for hypersurfaces with constant shifted curvature functions in warped product manifolds.
The study of multisymplectic structures using Spencer cohomology.
problem Integrability of multisymplectic structures.
method Applying Spencer cohomology to identify multisymplectic structures as G-structures and giving conditions for integrability. result Conditions for a multisymplectic form to admit a chart with constant coefficients.
Study on singularities of discrete Weingarten surfaces in Riemannian and Lorentzian spaceforms.
problem Analysis of singularities in discrete Weingarten surfaces.
method Defined and analyzed singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in 3D Riemannian and Lorentzian spaceforms.
result Discussed singularities of discrete surfaces with non-zero constant Gaussian curvature and parallel surfaces of discrete minimal and maximal surfaces.
We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described explicitly, proving that they are not always embedded. Besides new examples of Dela…
Researchers prove stability of black holes with positive cosmological constant.
problem Stability of black holes with positive cosmological constant.
method Developed a general framework to handle diffeomorphism invariance, used iteration scheme to solve Einstein's equations.
result Established global non-linear stability of Kerr-de Sitter black holes.
We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on 3-manifolds with positive Ricci curvature. In our estimate we make no assumption on the …
We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably \emph{improves} with the data size, requiring an essentially constant number of…
Unified method for solving extrinsic geometry problems.
problem General equivalence problem of extrinsic geometry.
method Formulation of osculating maps and algorithm for invariants.
result Categorical isomorphism between extrinsic geometries and involutive systems of linear differential equations.
Piecewise-linear regression trees improve tree-based regression with theoretical and practical benefits.
problem Improving tree-based regression models with theoretical guarantees and practical tractability.
method Regularized piecewise-linear node-splitting criterion, LASSO-type and ℓ2 regularization, variable selection procedure. result New high-probability generalization error bounds for piecewise-linear regression trees.
Researchers found a new type of singularity in surface evolution equations.
problem Finite-time singularity formation in surface evolution equations.
method Constructed first example of finite time blow-up solutions for the heat flow of the H-system.
result Singularity forms as a scaled least energy H-bubble with decoupled linearized operators.
Constructs a solution operator for hyperbolic gluing in higher dimensions.
problem Solving the linearized constant scalar curvature equation at hyperbolic space.
method Constructs a solution operator with good support propagation properties.
result Extends the Mao-Oh-Tao gluing method to the hyperbolic setting, gaining two derivatives.
The paper develops theory for holomorphic null curves in SL2(C).
problem Developing theory for holomorphic null curves in SL2(C).
method Establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for holomorphic null immersions.
result Proves every open Riemann surface admits a proper holomorphic null embedding into SL2(C).
Five types of translation surfaces with constant curvature found in Galilean 3-space.
problem Identifying translation surfaces with constant curvature in Galilean 3-space.
method Examining five types of translation surfaces based on the planarity of translating curves and absolute figure.
result Constant curvature translation surfaces with arbitrary Gaussian and mean curvature are found.
New Finsler metrics with constant flag curvature defined using Weyl-type curvature tensor.
problem Characterizing Finsler metrics with constant flag curvature.
method Defining a Weyl-type curvature tensor and constructing projectively related Finsler metrics.
result Construction of new families of Finsler metrics with constant flag curvature.
Geodesic flows on surfaces have specific fractional-linear integrals related to constant cross-ratios.
problem Characterizing geodesic flows on surfaces with fractional-linear integrals.
method Proving the dimension of fractional-linear integrals and giving a geometric criterion.
result The dimension of fractional-linear integrals is either 3 or 5, corresponding to constant curvature.
Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.
problem Creating Kähler metrics with constant scalar curvature on complex manifolds.
method Blowing up the manifold at points and constructing metrics with Poincaré-type singularities.
result Existence of constant scalar curvature Kähler metrics with Poincaré-type singularities.
Solves geodesic equations on specific metrics types.
problem Finding explicit solutions for geodesic equations on Eguchi-Hanson metrics.
method Analyzes geodesic equations under different constant conditions.
result Explicit solutions not yet available for geodesic equations when only ψ is constant. We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal …
New structures on symplectic manifolds derived from convex functions and matrices.
problem Investigating new types of toric generalized Kaehler structures on compact manifolds.
method Characterizing structures by triples (τ,C,F), proving canonical structures, and showing reversibility. result Underlying each structure is a canonical toric Kähler structure with a symplectic potential given by τ. Note on Einstein-type manifolds with constant curvature.
problem Characterizing Einstein-type manifolds with constant scalar curvature.
method Analyzing properties of gradient Einstein-type manifolds, including compact and noncompact cases.
result Nontrivial gradient Einstein-type manifolds are isometric to the standard sphere.
Termination proof for Cartan's method in constant type problems.
problem Proving termination of Cartan's equivalence method for constant type problems.
method Groupoid approach to Lie pseudo-groups and Cartan-Kuranishi theorem.
result Cartan's method terminates at involution or complete reduction for constant type problems.
Generalizes Escobar-Riemann mapping problem for smooth metric measure spaces.
problem Finding a function that attains the Escobar weighted constant.
method Introducing Escobar quotient, infimum, and resolving the problem when the weighted constant is negative.
result Obtained an Aubin type inequality connecting weighted Escobar constant and optimal constant for trace inequality.
Classifies surfaces with constant mean curvature in a specific type of space.
problem Classifying surfaces with constant mean curvature in a specific type of space.
method Classification based on the type of curvature (elliptic, hyperbolic, parabolic) and the rotational nature of the surfaces.
result Classification of constant mean curvature rotational surfaces.
The study explores metrics with constant curvature on compact manifolds.
problem Finding Hermitian metrics with constant second scalar curvature on compact manifolds.
method Analyzes Yamabe-type and elliptic equations, derives geometric consequences, and proves existence under specific curvature conditions.
result Under certain curvature conditions, a pluriclosed Gauduchon Hermitian metric has constant second Chern scalar curvature, leading to the existence of Kähler-Einstein metrics.
Study rigidifies Einstein-type manifolds with boundary and constant curvature.
problem Classifying compact Einstein-type manifolds with boundary and constant scalar curvature.
method Applied recent results on gradient Einstein-type manifolds to prove rigidity.
result Rigidity results for compact Einstein-type manifolds with boundary and constant scalar curvature.
The paper explores Finsler-type objects and their variational problems on spacetimes.
problem Generalizing Einstein equations to the Finsler setting.
method Study of the ladder of Finsler-type objects and their variational problems.
result Application of the ladder structure to variational proposals for Finsler spacetimes.
We prove that the isoperimetric constant is positive for all symmetric spaces of noncompact type and compute it explicitly.
Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.
problem Creating metrics with negative scalar curvature.
method Constructed two types of Eguchi-Hanson metrics.
result Found metrics with negative scalar curvature.
Paper investigates Lipschitz constants of self-attention modules in neural networks.
problem Lipschitz constants of self-attention modules in neural networks.
method Proved standard dot-product self-attention is not Lipschitz for unbounded input domain. Proposed L2 self-attention that is Lipschitz. Derived upper bound on L2 self-attention's Lipschitz constant.
result Proved standard self-attention is not Lipschitz for unbounded input domain and proposed an alternative L2 self-attention that is Lipschitz.
We study the asymptotic behavior of the sequence of the Nielsen numbers {N(fk)}, the essential periodic orbits of f and the homotopy minimal periods of f by using the Nielsen theory of maps f on infra-solvmanifolds of type R. We give a linear lower bound for the number of essential periodic orbits of such …
In this paper we prove a variation of the theorem in title, for equations with periodic coefficients, in Frechet spaces. The main result gives equivalent conditions ensuring the reduction of such an equation to one with constant coefficient. In the particular case of C∞, we obtain the exact analogue of the cl…
For a two-dimensional surface in the four-dimensional Euclidean space we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and kappa. The condition k = kappa = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are charac…
New method solves large-scale linear programming problems for sparse signal reconstruction.
problem Efficiently solving large-scale linear programming problems for sparse signal reconstruction.
method Combining constraint and column generation techniques with simplex method initialization.
result Highly efficient solutions for many settings.
We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this …
Estimates population profile from small random samples.
problem Learning population composition from limited data.
method Minimum distance estimator based on linear programming.
result Consistent estimation of profile in sublinear sample size.
Finsler metrics of constant curvature characterized, with a Finslerian Beltrami Theorem.
problem Characterizing Finsler metrics of constant curvature.
method Defined a Weyl-type curvature tensor to characterize Finsler metrics of constant flag curvature.
result Provided a Finslerian version of Beltrami Theorem.
In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.
Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.
problem Estimating eigenvalues and spectrum for graph substructures.
method Introducing Cheeger type constants via isocapacitary constants to estimate eigenvalues and spectrum.
result Estimates for first Dirichlet, Neumann, and Steklov eigenvalues, as well as the bottom of the spectrum of the Laplace operator and Dirichlet-to-Neumann operator.
In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescal…