Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
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.
This paper studies ruled real hypersurfaces in indefinite complex projective space.
problem Characterizing and classifying ruled real hypersurfaces in indefinite complex projective space.
method Introduced and studied ruled real hypersurfaces with maximal holomorphic distribution integrable and leaves totally geodesic holomorphic hyperplanes. Detailed shape operator computation and method of construction by gluing totally geodesic hyperplanes along a curve.
result Classification of all minimal ruled real hypersurfaces in terms of three main families of curves.
In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the inv…
Proves a limit on hyperplanes in complex manifolds.
problem Limiting the number of hyperplanes in complex manifolds.
method Effective density theorem for periodic orbits, Margulis functions, restricted projection theorem, equidistribution result.
result Proves a quantitative finiteness theorem for hyperplanes.
Study shows only hyperplanes in Heisenberg groups have zero curvature.
problem Understanding Bernstein problem in higher dimensional Heisenberg groups.
method Sub-Riemannian characterization of ruling property and study of geodesics.
result Only hyperplanes have zero horizontal symmetric second fundamental form in Heisenberg groups.
Study proves a new formula for capillary hypersurfaces and shows a flow converging to a special shape.
problem Understanding the behavior of capillary hypersurfaces in hyperbolic space.
method Developed a volume-preserving flow starting from a star-shaped initial hypersurface and proved its long-time existence and convergence.
result The flow converges to a θ-totally umbilical cap, which is an energy minimizer for a given enclosed volume. The paper examines stable capillary hypersurfaces in hyperbolic space.
problem Stability of capillary hypersurfaces with free boundary on a horosphere.
method Analysis of umbilical and totally geodesic hypersurfaces using stability criteria.
result Umbilical and totally geodesic hypersurfaces are the only stable capillary hypersurfaces with boundary on a horosphere.
The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a…
Study on minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.
problem Behavior of minimal hypersurfaces in Schwarzschild manifolds intersecting the horizon orthogonally.
method Analysis of free boundary minimal hypersurfaces and totally geodesic hyperplanes in Schwarzschild n-manifolds. result A free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region.
New method constructs asymptotic convex hypersurfaces via equidistant hyperplanes.
problem Constructing asymptotic convex hypersurfaces in hyperbolic space.
method Approximating hypersurface by geodesic graphs over equidistant hyperplanes.
result Existence of complete, strictly locally convex hypersurfaces with prescribed asymptotic boundary.
We consider geodesic flows between hypersurfaces in Rn. However, rather than consider using geodesics in Rn, which are straight lines, we consider an induced flow using geodesics between the tangent spaces of the hypersurfaces viewed as affine hyperplanes. For naturality, we want the geodesic flow to be invaria…
We prove an explicit residue formula for a meromorphic continuation of conformally covariant integral operators between differential forms on Rn and on its hyperplane. The results provide a simple and new construction of the conformally covariant differential symmetry breaking operators between differential fo…
New hyperbolic 3-pseudomanifolds with unique properties.
problem Understanding cubulable groups in hyperbolic 3-manifolds.
method Constructing compact hyperbolic 3-manifolds with specific boundary conditions.
result Found groups that are word hyperbolic but not cubulable.
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually…
In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in Rn+1(n=2,3) is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton Σ with nonnegative scalar curva…
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
We study spaces obtained from a complete finite volume complex hyperbolic n-manifold M by removing a compact totally geodesic complex (n-1)-submanifold. The main result is that the fundamental group of M-S is relatively hyperbolic, relative to fundamental groups of the ends of M-S, and M-S admits a complete finite volu…
Classifies totally geodesic submanifolds in symmetric spaces.
problem Classifying submanifolds in symmetric spaces.
method Classification of totally geodesic submanifolds in products of rank one symmetric spaces.
result Infinitely many examples of irreducible totally geodesic submanifolds in Hermitian symmetric spaces.
A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curve…
We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.
Totally geodesic surfaces found in knots and links.
problem Finding totally geodesic surfaces in knots and links.
method Constructing infinite families of knots and links with totally geodesic spanning surfaces in various 3-manifolds.
result Infinite families of knots and links with totally geodesic spanning surfaces in multiple 3-manifolds.
Finite totally geodesic hypersurfaces in curved manifolds proven.
problem Characterizing totally geodesic hypersurfaces in curved manifolds.
method Analytic Riemannian manifold analysis with negative sectional curvature.
result Closed manifolds with negative curvature have only finitely many totally geodesic hypersurfaces.
Classifies totally geodesic submanifolds in exceptional symmetric spaces.
problem Classifying totally geodesic submanifolds in exceptional symmetric spaces.
method Classification and introduction of an invariant (Dynkin index) for totally geodesic embeddings.
result Existence of a totally geodesic submanifold of minimal codimension with specific properties.
The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.
problem Classifying hypersurfaces in quaternionic space forms with specific curvature properties.
method Analyzing curvature-adapted real hypersurfaces in non-flat quaternionic space forms HPm and HHm. result Classification of hypersurfaces including geodesic hyperspheres, tubes, and specific examples in HPm and HHm. Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
Classifies totally geodesic submanifolds in specific geometric spaces.
problem Identifying totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
method Developed new techniques for studying totally geodesic submanifolds in analytic Riemannian manifolds, homogeneous spaces, and Riemannian cones.
result Obtained a classification of totally geodesic submanifolds in homogeneous nearly Kähler 6-manifolds and their G2-cones.
Paper proves Fujimoto's conjecture for even m ≥ 4.
problem Proving Fujimoto's conjecture for even dimensions.
method Using a special planar network in the theory of positive matrices.
result Best possible number for even m ≥ 4 is proven.
The study finds totally geodesic surfaces in hyperbolic 3-manifolds and verifies a conjecture.
problem Finding totally geodesic surfaces in hyperbolic 3-manifolds.
method Developed algorithms to determine and verify the existence of totally geodesic surfaces.
result Discovered nine 3-manifolds with totally geodesic surfaces and verified Menasco-Reid's conjecture for knots up to 12 crossings.
Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.
problem Characterizing minimal hypersurfaces in S^5 with certain curvature conditions.
method Analyzing hypersurfaces with constant scalar curvature and zero Gauss curvature.
result Minimal hypersurfaces in S^5 with these curvature properties are totally geodesic.
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
Totally geodesic Lagrangian submanifolds in nearly Kähler S³×S³.
problem Characterizing Lagrangian submanifolds in nearly Kähler manifolds.
method Analyzing H-umbilical properties and their implications for geodesicity. result In nearly Kähler S³×S³, H-umbilical Lagrangian submanifolds are totally geodesic. The first examples of totally geodesic Seifert surfaces are constructed for hyperbolic knots and links, including both free and totally knotted surfaces. Then it is proved that two bridge knot complements cannot contain totally geodesic orientable surfaces.
A metric Lie algebra g is a Lie algebra equipped with an inner product. A subalgebra h of a metric Lie algebra g is said to be totally geodesic if the Lie subgroup corresponding to h is a totally geodesic submanifold relative to the left-invariant Riemannian metric defined by the inner product, on the simply connected …
One purpose of this article is to establish a general method to determine stability of totally geodesic submanifolds of symmetric spaces. The method is used to determine the stability of the basic totally geodesic submanifolds M+,M− introduced and studied by Chen and Nagano in [Totally geodesic submanifolds of symm…
Study counts geodesic surfaces in knot complements, finding unique ones for small knots.
problem Counting totally geodesic surfaces in knot complements.
method Adapting boundary slope and intersection techniques, extending obstructions.
result Uniqueness of geodesic surfaces for specific knots, no geodesic surfaces for 47 knots.
Classifies special hypersurfaces in Gödel spacetimes.
problem Characterizing hypersurfaces in Gödel spacetimes.
method Classification of parallel and totally geodesic hypersurfaces.
result Identified specific types of hypersurfaces in Gödel spacetimes.
This survey is an introduction to the geometry of co-Minkowksi space, the space of unoriented spacelike hyperplanes of the Minkowski space. Affine deformations of cocompact lattices of hyperbolic isometries act on it, in a way similar to the way that quasi-Fuchsian groups act on hyperbolic space. In particular, there i…
Totally geodesic subvarieties in moduli spaces are studied.
problem Characterizing totally geodesic subvarieties in moduli spaces.
method Analyzing the Deligne-Mumford boundary and its strata.
result Boundary loci of totally geodesic subvarieties are themselves totally geodesic.
Study on embedding hyperbolic 2-orbifolds in Bianchi orbifolds.
problem Embedding closed totally geodesic hyperbolic 2-orbifolds in Bianchi orbifolds.
method Analyzing Bianchi orbifolds H3/PSL(2,Od) for large d. result Existence of at least cd closed embedded totally geodesic hyperbolic 2-orbifolds for large d. Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gauß map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.
Study shows certain 4D hyperbolic links don't contain geodesic 3-manifolds.
problem Proving certain hyperbolic link complements don't contain geodesic 3-manifolds.
method Analyzing hyperbolic link complements of 2-tori in S^4.
result Proves certain hyperbolic link complements do not contain closed embedded totally geodesic hyperbolic 3-manifolds.
In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Rieman…
We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.
Totally geodesic dual leaves on curved manifolds are also curved.
problem Characterizing dual leaves of nonnegatively curved polar manifolds.
method Proving dual leaves are totally geodesic and closed, and inducing a Riemannian submersion.
result Dual leaves of nonnegatively curved polar manifolds are themselves nonnegatively curved and totally geodesic.
Study totally umbilic submanifolds using planar pseudo-geodesics.
problem Characterize totally umbilic isometric immersions with parallel normalized mean curvature vector.
method Introduce planar pseudo-geodesics and analyze their properties; prove the equivalence of totally umbilic immersions and planar geodesic extrinsic shapes.
result An isometric immersion is totally umbilic if and only if every geodesic of the manifold has planar extrinsic shape.
Study extends geodesic curvature formula to higher dimensions.
problem Extending curvature formula to higher-dimensional spheres.
method Using new integral-geometric formulas for Euclidean and geodesic total curvature.
result Explicit formula for geodesic total curvature on higher-dimensional spheres.
The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of total…