Estimates heights of special surfaces in warped products.
problem Estimating heights of special surfaces in warped products.
method Provided a vertical height estimate.
result Vertical height estimates for special Weingarten surfaces of elliptic type in warped products.
The paper modifies a warped product space to find conditions for constant height functions.
problem Finding sufficient conditions for the height function to be constant in a modified warped product space.
method The paper modifies the warped product space by adding a warping function and discusses the sufficient condition for the height of immersed surfaces.
result The paper establishes a sufficient condition for the height function to be constant in the modified warped product space.
Geometrically describes surfaces with parallel mean curvature in warped product spaces.
problem Understanding surfaces with parallel mean curvature in warped product spaces.
method Obtained a geometric description using the normal connection.
result Extended a result by Alencar-do Carmo-Tribuzy on surfaces with parallel mean curvature.
The study classifies totally umbilical surfaces in warped product manifolds.
problem Classifying totally umbilical surfaces in warped product manifolds.
method Analyzing surfaces in warped product M(κ)fimesI and finding first integrals. result Totally umbilical surfaces in M(κ)fimesI are invariant by isometries. Flow preserves area, proving a general isoperimetric inequality.
problem Proving a general isoperimetric inequality in warped product surfaces.
method Mean curvature flow in warped product surfaces, preserving area.
result General isoperimetric inequality for radial graphs in warped product surfaces.
The paper derives height estimates for surfaces with constant curvature in warped product spaces.
problem Estimating heights of surfaces with constant curvature in warped product spaces.
method Use of conformal parameters and geometric applications to derive height estimates.
result Derives height estimates for surfaces with positive extrinsic or mean curvature in RimesfR2. The paper estimates heights of H-surfaces in warped products with conditions.
problem Estimating heights of constant mean curvature surfaces in warped products.
method Analyzing surfaces with transversal boundaries in warped product spaces.
result Height estimates involving area and bounded volume for H-surfaces.
The paper characterizes biharmonic submersions from product manifolds.
problem Characterizing biharmonic Riemannian submersions from M2imesR. method Local characterizations and by-products of biharmonic submersions.
result Local characterizations of biharmonic Riemannian submersions and uniqueness of a specific submersion.
We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, …
The study classifies warped products with harmonic curvature on surfaces, showing two possibilities for the metric.
problem Classifying warped products with harmonic curvature on surfaces.
method Analyzing the properties of nonconstant warping functions and Gaussian curvature.
result Both possibilities of metrics are realized on closed orientable surfaces of genus greater than 1.
Investigates conditions for existence of inverse mean curvature flow in warped product manifolds.
problem Ensuring existence of inverse mean curvature flow in warped product manifolds.
method Analyzes sufficient curvature conditions for the Riemannian metric.
result Determines conditions on curvature for flow existence.
Sharp upper bounds found for Steklov eigenvalues of warped products.
problem Finding bounds for Steklov eigenvalues of specific metric configurations.
method Investigation of Steklov spectrum for warped products with a fiber of dimension 2.
result Sharp upper bounds for Steklov eigenvalues in terms of the eigenvalues of the Laplacian on the fiber.
Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.
problem Characterize elliptic Weingarten surfaces in warped product spaces with minimal type curvature conditions.
method Analyze surfaces with mean curvature and extrinsic curvature satisfying a specific relationship under radial symmetry of the warping function.
result Existence and uniqueness of rotationally-invariant elliptic Weingarten surfaces of minimal type in RimeshR. The paper proves inequalities and formulas for submanifolds in specific warped product manifolds.
problem Understanding geometric properties of submanifolds in warped product manifolds.
method Proving linear isoperimetric inequalities and monotonicity formulas for submanifolds with bounded mean curvature vector.
result Lower bound estimates for the volume of submanifolds in terms of the warping function.
New types of Einstein manifolds discovered from homogeneous surfaces.
problem Understanding quasi-Einstein equations on homogeneous surfaces.
method Modified Riemannian extension and warped product construction.
result Discovery of new Einstein manifolds.
We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not tot…
Let I⊆R be an open interval, f:I→R a strictly positive function and denote by $\E^2$ the Euclidean plane. We classify all surfaces in the warped product manifold $I \times_f \E^2$ for which the unit normal makes a constant angle with the direction tangent to I.
Study Lagrangian immersions in nearly Kähler S^6, finding special cases.
problem Characterizing Lagrangian immersions in nearly Kähler S^6.
method Warped product approach, analyzing sectional curvature and Chen's inequality.
result Special Lagrangian immersions have constant sectional curvature 1/16 or satisfy Chen's equality.
The paper studies curve shortening flows on non-convex surfaces.
problem Behavior of curve shortening flows on non-convex surfaces.
method Defined a graph property and proved its preservation under curve shortening flow.
result The curve becomes a graph after a finite time under the curve shortening flow.
Study on four-dimensional Ricci solitons and multiply warped Ricci flow solutions.
problem Characterizing singularities in Ricci flow solutions.
method Classification and rigorous examples of multiply warped Ricci flow solutions.
result Formation of generalized cylinder singularity models.
Study of mean curvature flow in Fuchsian manifolds, proving convergence for certain initial surfaces.
problem Detecting minimal surfaces in hyperbolic manifolds.
method Investigation of mean curvature flow in Fuchsian manifolds, proving existence and convergence for specific initial surfaces.
result Existence and convergence of mean curvature flow for certain initial surfaces in Fuchsian manifolds.
The paper proves compactness of warped product metrics on S²×S¹ with varying base metrics.
problem Proving compactness of warped product metrics on S²×S¹ with nonnegative scalar curvature.
method Using Gromov-Sormani MinA scalar curvature compactness conjecture, the paper proves a uniform diameter bound for the base surfaces, compactness of the base warping functions, and convergence of the metrics.
result The metrics converge to a limit metric with nonnegative scalar curvature in the distributional sense.
The paper proves conditions for stable constant mean curvature surfaces in specific manifolds.
problem Conditions for stable constant mean curvature surfaces in warped product manifolds.
method Analyzes de Sitter-Schwarzschild and Reissner-Nordstrom manifolds, then generalizes to a broader class of three-dimensional warped product manifolds.
result Stable, compact surfaces in specific manifolds are embedded topological spheres.
Cylinders in warped product spaces have zero curvature.
problem Characterizing cylinders in warped product spaces.
method Proving cylinders have zero extrinsic and intrinsic curvatures.
result Cylinders in M2imesRn have zero curvature. Study investigates minimal surfaces in spherical caps, extending previous findings.
problem Characterizing minimal surfaces with free boundaries and capillary conditions in spherical caps.
method Extending previous half-space intersection properties to warped products and capillary minimal surfaces in high codimension.
result Established a dual operation relating free boundary and capillary minimal surfaces.
Study solutions to affine quasi-Einstein equation on homogeneous surfaces.
problem Understanding solutions to affine quasi-Einstein equation on homogeneous surfaces.
method Using the dimension of affine Killing vector fields as a framework, we provide explicit descriptions of solution spaces.
result Very explicit descriptions of solution spaces for gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds.
The article surveys recent results on warped and CR-warped product submanifolds in Kaehler manifolds.
problem Characterizing submanifolds in Kaehler manifolds using warped and CR-warped products.
method Analyzing properties of warped and CR-warped products in the context of Kaehler manifolds.
result Presented several results on the geometry of warped and CR-warped product submanifolds.
The paper examines Einstein doubly warped product manifolds with a semi-symmetric metric connection.
problem Characterizing Einstein doubly warped product manifolds with a semi-symmetric metric connection.
method Deriving curvature formulas and proving necessary and sufficient conditions for a manifold to be a warped product.
result Obtained results for Einstein doubly warped product manifolds and Einstein-like doubly warped product manifolds.
Generalizes warped product submersion to conformal case.
problem Understanding angles preservation in submersions.
method Introduces conformal warped product submersion.
result Fundamental tensors derived for conformal submersion.
Study compact sequences of warped product circles over spheres with nonnegative scalar curvature.
problem Compactness of sequences of warped product circles over spheres with nonnegative scalar curvature.
method Proved subsequence convergence to a W1,p Riemannian metric for all p<2 with nonnegative scalar curvature in the distributional sense. result Proved compactness of sequences of warped product circles over spheres with nonnegative scalar curvature.
Smooth convergence to a sphere from cone-shaped surfaces.
problem Evolution of surfaces inside a cone using inverse mean curvature flow.
method Inverse mean curvature flow applied to star-shaped hypersurfaces meeting a convex cone perpendicularly.
result Smooth convergence to a sphere over time.
We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density. We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such…
The paper explores geometric properties of Riemannian warped product maps and their curvature.
problem Investigating the geometric properties of Riemannian warped product maps.
method The approach involves establishing conditions for geodesics, deriving curvature tensors, and examining various types of maps.
result Derivation of integral formula for scalar curvature of conformal Riemannian warped product maps.
Study on warped product Yamabe solitons with constant fiber curvature.
problem Characterizing nontrivial warped product Yamabe gradient solitons.
method Investigation of warped product manifolds, derivation of scalar curvature estimates.
result Nontrivial warped product Yamabe gradient solitons have constant scalar curvature in the fiber.
The paper explores warped-like product metrics with exceptional holonomy groups.
problem Exploring new metrics with exceptional holonomy groups.
method Presented a general ansatz of warped-like product metric and studied specific examples.
result Explicit examples of (3+3+2) and (3+3+1) warped-like product manifolds with Spin(7) and G2 holonomy, respectively. Proves a theorem for a specific type of warped product space.
problem Analyzing warped product spaces.
method Uses the warped function \( f(r) = \cosh\left(r\sqrt{\frac{\lambda}{n-2}}
ight) \) to prove the theorem.
result Proves an almost splitting theorem for the specified warped product space.
Hopf's theorem generalized to curved spaces.
problem Extending Hopf's theorem to non-Euclidean spaces.
method Curvature flows and warped product manifolds.
result Generalized Hopf's theorem to specific curved spaces.
Warped product affects divergences in information geometry.
problem Warped product's impact on divergences in information geometry.
method Study of warped product on information geometry.
result Warped product does not preserve canonical divergences.
The paper extends affine connection results to singular warped and twisted products.
problem Generalizing affine connections to singular warped and twisted products.
method Study of singular multiply warped products and singular twisted products with semi-symmetric metric and non-metric connections, discussing Koszul forms and curvature.
result Theoretical results on curvature and Koszul forms for singular multiply warped and twisted products.
The paper proves rigidity for submanifolds in warped product manifolds.
problem Rigidity of submanifolds in warped product manifolds.
method Dihedral extremality and rigidity theorem for submanifolds with polyhedral boundary.
result Dihedral rigidity results for hyperbolic polyhedra in flat warped product spaces.
Paper defines and studies Clairaut warped product Riemannian maps.
problem Understanding the geometry of specific Riemannian maps.
method Identify geodesic conditions, derive conditions for Clairaut maps, and calculate curvature.
result Found conditions for a warped product Riemannian map to be Clairaut.
Study on warped product pointwise semi-slant submanifolds in Sasakian and cosymplectic manifolds.
problem Investigate the geometry of pointwise semi-slant submanifolds and their warped products.
method Using the notion of pointwise slant submanifolds, investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds.
result Prove the existence of no proper pointwise semi-slant warped product submanifolds other than contact CR-warped products in Sasakian manifolds.
The paper studies a new type of submanifolds in product spaces.
problem Characterizing and understanding warped product pointwise bi-slant submanifolds.
method Introduced and studied warped product pointwise bi-slant submanifolds of locally product Riemannian manifolds.
result Characterization results and non-trivial examples of these submanifolds.
Study properties of bi-warped product submanifolds in specific geometric spaces.
problem Characterize geometric properties of bi-warped product submanifolds.
method Analyze squared norm of second fundamental form and warping functions.
result Relationship between squared norm and warping functions for proper slant submanifolds.
Study shows how certain metrics can be split into warped products.
problem Understanding conditions under which metrics can be split as warped products.
method Investigating warped area-minimizing hypersurfaces and spectral Ricci/scalar curvature bounds.
result Metrics can be locally split as warped products under specific curvature conditions.
Characterizes and examines gradient solitons on doubly warped product manifolds.
problem Understanding gradient solitons on specific manifold structures.
method Characterizations and examinations of various types of gradient solitons on doubly warped product manifolds.
result Effects of gradient solitons on factor manifolds and specific curvature properties of doubly warped products.
The paper characterizes warped product submanifolds in LCS-manifolds.
problem Characterizing warped product submanifolds in LCS-manifolds.
method Analyzing contact CR-warped product and pseudo-slant submanifolds, and constructing examples.
result There do not exist any proper warped product bi-slant submanifolds of LCS-manifolds.
New findings on instability of Einstein metrics in warped products.
problem Investigating the stability of warped product Einstein metrics.
method Exploiting the relationship between warped product Einstein metrics, quasi-Einstein metrics, and Ricci solitons, introducing a new destabilising perturbation (the Ricci variation).
result Certain infinite families of warped product Einstein metrics are unstable in high dimensions.