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.
The paper studies how certain surfaces evolve in space-time.
problem Preserving the space-like condition of non-compact hypersurfaces.
method Prescribed mean curvature flow in generalized Robertson-Walker spaces.
result The flow preserves space-like condition and exists for infinite time.
Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.
problem Evolution of non-compact convex hypersurfaces in Rn+1 by inverse mean curvature. method Establish long-term existence via pointwise mean curvature estimate and viscosity solutions for strict convexity.
result Characterization of maximal time of existence in terms of tangent cone at infinity.
Proves well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
problem Proving well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.
method Analyzes globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces.
result Proves well-posedness of the Cauchy problem for the Dirac operator.
We show that the space of min-max minimal hypersurfaces is non-compact when the manifold has an analytic metric of positive Ricci curvature and dimension 3≤n+1≤7. Furthermore, we show that bumpy metrics with positive Ricci curvature admit minimal hypersurfaces with unbounded index+area. When combined with the…
Minimal surfaces found in finite volume manifolds.
problem Existence of minimal hypersurfaces in complete manifolds of finite volume.
method Independent result on volume sweeping by hypersurfaces; main tool is a volume constraint result.
result Proves existence of minimal hypersurfaces in complete non-compact manifolds of finite volume.
In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
The paper finds closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
problem Finding closed hypersurfaces with prescribed mean curvature in non-compact manifolds.
method One-parameter prescribed mean curvature min-max theory.
result Closed hypersurfaces with prescribed mean curvature are found in certain non-compact manifolds.
The paper proves inequalities for manifolds with boundary and non-compact regions.
problem Analyzing the capacity and rigidity of manifolds with boundary.
method Inverse mean curvature flow for hypersurfaces with boundary.
result Proves inequalities involving total mean curvature and mass.
In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For doing this, we develop a modified min-max theory for the area functional following…
Extends Atiyah-Singer Dirac operator study to non-compact spacetimes.
problem Analyzing Dirac operator on non-compact spacetimes with non-compact Cauchy hypersurface.
method Building on previous works, extends Fredholm result to non-compact Lorentzian spaces, using von Neumann algebras and Galois coverings.
result Γ-Fredholmness of the Dirac operator under APS boundary conditions.
The paper studies stability and area growth of λ-hypersurfaces.
problem Stability and growth of area for λ-hypersurfaces. method Defined a F-functional and studied F-stability. result Lower and upper bounds for area growth of λ-hypersurfaces. Static potentials on flat manifolds imply zero boundary values or non-compact area minimizers.
problem Characterizing static potentials on asymptotically flat manifolds.
method Analyzing the properties of static potentials and area minimizing hypersurfaces.
result Asymptotically flat manifolds with static potentials either have zero boundary values or contain non-compact area minimizers.
Researchers extend Gamma index theorem to non-compact spacetimes.
problem Establishing an L2-Gamma index for non-compact spacetimes. method Rewriting L2-Gamma index in terms of spectral flow and connecting to geometric expressions. result Extends Bär and Strohmaier's work to non-compact Cauchy hypersurfaces.
We prove some new rigidity results for proper biharmonic immersions in Sn of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
The paper shows that immersions of hypersurfaces can fail to be manifolds and converges under certain conditions.
problem Non-compactness of W2,d immersions of d-dimensional hypersurfaces. method Construction of a family of immersions and analysis of their limits.
result A family of immersions whose limit fails to be an immersion of a manifold.
Study on free boundary minimal hypersurfaces in Schwarzschild space, proving zero Morse index for certain hypersurfaces.
problem Analyzing free boundary minimal hypersurfaces in the Riemannian Schwarzschild space.
method Variational methods and geometric analysis.
result Zero Morse index for certain free boundary rotationally symmetric totally geodesic hypersurfaces in the Riemannian Schwarzschild space.
Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
Proves a Liouville-type theorem for p-Laplacian on manifolds.
problem Proving Liouville-type theorems for p-Laplacian on manifolds.
method Proved a Liouville-type result for the p-Laplacian on complete Riemannian manifolds.
result Proved a Liouville-type theorem for the p-Laplacian on complete non-compact Riemannian manifolds.
Here are described the geometric structures of the lines of principal curvature and the partially umbilic singularities of the tridimensional non compact generic quadric hypersurfaces of R4. This includes the ellipsoidal hyperboloids of one and two sheets and the toroidal hyperboloids. The present study co…
Obstructs complete metrics with positive scalar curvature on non-compact manifolds.
problem Obstructing complete metrics with positive scalar curvature on non-compact manifolds.
method Using minimal hypersurfaces and MOTS, the study provides topological obstructions and proves the Liouville theorem.
result The Liouville theorem for locally conformally flat n-manifolds of non-negative scalar curvature follows from the impossibility of positive scalar curvature metrics.
We investigate Inverse Mean Curvature Flow (IMCF) of non-compact hypersurfaces in hyperbolic space. Specifically, we look at bounded graphs over horospheres in Hn+1 and show long time existence of the flow. Along the way many important local estimates as well as global estimates are obtained. In addition,…
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
The paper proves a positive mass theorem for non-compact static domains in hyperbolic space.
problem Proving a positive mass theorem for non-compact static domains in hyperbolic space.
method Formulating and proving a positive mass theorem under natural dominant energy conditions, using elliptic boundary conditions on spinors.
result Retrieve a sharper version of a recent result by Souam about the rigidity of non-compact static domains.
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
In this paper, we introduce a definition of λ-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that λ-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete λ-hypersurfaces with …
New findings on stable minimal hypersurfaces in curved 4-manifolds.
problem Nonexistence of complete stable minimal hypersurfaces in positively curved 4-manifolds.
method Combination of non-negative sectional curvature and strict positivity of scalar curvature.
result Rigidity of complete stable minimal hypersurfaces in 4-manifolds with positive curvature.
Flow of curved surfaces converges to a specific shape over time.
problem Behavior of curved surfaces over time.
method Addressed through the α-Gauss curvature flow for α>1/2. result Flow converges to a translating soliton determined by initial conditions.
The paper proves residue formulas for logarithmic foliations on non-compact manifolds.
problem Analyzing logarithmic foliations on non-compact complex manifolds.
method Proves Baum-Bott type formula for residue.
result Provides a Poincaré-Hopf type theorem and optimal description for foliations.
New examples of real hypersurfaces found in complex hyperbolic quadrics.
problem Existence of specific types of real hypersurfaces in complex hyperbolic quadrics.
method Construction of a one-parameter family of homogeneous Hopf hypersurfaces.
result First known examples of real hypersurfaces with integrable maximal complex subbundle in irreducible Kahler manifolds.
We show that closed hypersurfaces in Euclidean space with nonnegative scalar curvature are weakly mean convex. In contrast, the statement is no longer true if the scalar curvature is replaced by the k-th mean curvature, for k greater than 2, as we construct the counter-examples for all k greater than 2. Our proof relie…
In this paper, we study rigidity problems for hypersurfaces with constant curvature quotients H2kH2k+1 in the warped product manifolds. Here H2k is the k-th Gauss-Bonnet curvature and H2k+1 arises from the first variation of the total integration of $…
Minimal hypersurfaces scarring along a fixed one in certain manifolds.
problem Scarring of minimal hypersurfaces in specific manifolds.
method Generic scarring phenomenon for minimal hypersurfaces in thick-at-infinity manifolds with thin foliation.
result Existence of sequences of minimal hypersurfaces scarring along a fixed one, with diverging area and renormalized convergence to the fixed hypersurface.
In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends Σn⊆Rn+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in $\math…
We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.
Study proves upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
problem Proving upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces.
method Analyzing a weighted eigenvalue problem and using a Lorentz-Sobolev inequality to study eigenfunctions and index/nullity in neck regions.
result Upper semicontinuity of index plus nullity for minimal and H-CMC hypersurfaces proved.
We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide severa…
Study on the index and Betti number of f-minimal hypersurfaces and self-shrinkers.
problem Estimating the Morse index of self-shrinkers and f-minimal hypersurfaces.
method Analyzing the relationship between the index and the first Betti number of compact hypersurfaces, and using the dimension of the space of weighted square summable f-harmonic 1-forms in the non-compact case.
result Lower bounds on the index in terms of the first Betti number and the dimension of the space of weighted square summable f-harmonic 1-forms.
Study finds rigid submanifolds in spacelike waves under specific conditions.
problem Understanding rigidity of submanifolds in spacelike waves.
method Proves submanifolds are contained in characteristic lightlike hypersurfaces under certain conditions.
result Complete codimension two submanifolds are wavefronts under specific conditions.
New isoparametric hypersurfaces found in Damek-Ricci spaces.
problem Characterizing new isoparametric hypersurfaces in Damek-Ricci spaces.
method Defining and studying 'sphere-like' hypersurfaces formed by extending horospheres.
result Found a new family of isoparametric hypersurfaces connecting geodesic spheres to previously known ones.
Let M be a Hadamard manifold with curvature bounded above by a negative constant −α, satisfying the "strict convexity condition", and assume that M admits a "helicoidal" one-parameter subgroup G of isometries of M. Then, given a compact topological G−shaped hypersurface Γ in the asymptotic boundary of $M,…
Willmore-type inequalities for bounded domains in manifolds with curvature bounds.
problem Establishing inequalities for bounded domains in manifolds with curvature bounds.
method Using asymptotic or integral Ricci curvature bounds to establish inequalities.
result Recovering a recent inequality of Jin-Yin.
Study inverse mean curvature flow in quaternionic hyperbolic space, proving flow properties and convergence.
problem Evolution of star-shaped hypersurfaces in quaternionic hyperbolic space.
method Inverse mean curvature flow, star-shaped hypersurface, mean convex, convergence analysis.
result Flow is defined for any positive time, evolving hypersurface stays star-shaped and mean convex, induced metric converges to a conformal multiple of the standard sub-Riemannian metric on the sphere.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
We study the geometry of families of hypersurfaces in Eguchi-Hanson space that arise as complex line bundles over curves in S2 and are three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf tori for closed curves. They are negatively curved, asymptotically flat spaces, and we compute the com…
The study of submanifolds with constant principal curvatures in symmetric spaces.
problem Characterizing submanifolds with constant principal curvatures in symmetric spaces.
method Systematic approach to constructing and classifying homogeneous submanifolds in irreducible Riemannian symmetric spaces of non-compact type.
result A large number of new examples of non-totally geodesic CPC submanifolds not coming from cohomogeneity one actions.