The study classifies hypersurfaces with constant principal curvatures in S 3 i m e s R \mathbb{S}^3 imes \mathbb{R} S 3 im es R and H 3 i m e s R \mathbb{H}^3 imes \mathbb{R} H 3 im es R .
problem Classifying hypersurfaces with constant principal curvatures in specific product spaces.
method Analyzing isoparametric surfaces and using isoparametric properties to classify hypersurfaces.
result Hypersurfaces with constant principal curvatures are cylinders over isoparametric surfaces in Q 3 \mathbb{Q}^3 Q 3 . In this paper, by using the G 2 G_2 G 2 -structure on Im ( O ) ≅ R 7 (\mathbb O)\cong\mathbb R^7 ( O ) ≅ R 7 from the octonions O \mathbb O O , the G 2 G_2 G 2 -binormal motion of curves γ ( t , s ) γ(t,s) γ ( t , s ) in R 7 \mathbb R^7 R 7 associated to the almost complex structure on S 6 \mathbb S^6 S 6 is studied. The motion is proved to be equivalent to Schrödinger flows from $\mathbb R^…
We define a notion of isotropic surfaces in O \mathbb{O} O , i.e. on which some canonical symplectic forms vanish. Using the cross-product in O \mathbb{O} O we define a map ρ : G r _ 2 ( O ) → S 6 ρ\colon Gr\_2(\mathbb{O})\to S^6 ρ : G r _2 ( O ) → S 6 from the Grassmannian of O \mathbb{O} O to S 6 S^6 S 6 . This allows us to associate to each surface Σ Σ Σ of O \mathbb{O} O a fun…
Two families of genus one surfaces with H H H -curvature in H 2 i m e s R \mathbb{H}^2 imes\mathbb{R} H 2 im es R are constructed.
problem Constructing surfaces with positive constant mean curvature in H 2 i m e s R \mathbb{H}^2 imes\mathbb{R} H 2 im es R . method Conjugate construction.
result There is no Schoen-type theorem for immersed surfaces with positive constant mean curvature in H 2 i m e s R \mathbb{H}^2 imes\mathbb{R} H 2 im es R . Special orthogonal representations from octonions have geometric properties linked to binary cubics.
problem Understanding geometric properties of special orthogonal representations from octonions.
method Using octonions and their derivations, spinors, and covariants to show geometric properties.
result Covariants and Mathews identities of these representations are related to the Fano plane and ( Z 2 ) 3 (\mathbb{Z}_2)^3 ( Z 2 ) 3 . The paper constructs surfaces with constant mean curvature in a specific space and explores their properties.
problem Existence and properties of surfaces with constant mean curvature in H 2 i m e s R \mathbb{H}^2 imes\mathbb{R} H 2 im es R . method Construction of a 2-parameter family of surfaces with specified symmetries and mean curvature bounds.
result The Krust property does not hold for surfaces with positive mean curvature in H 2 i m e s R \mathbb{H}^2 imes\mathbb{R} H 2 im es R . Classifies quotients of S n i m e s S n S^n imes S^n S n im es S n by Z / p i m e s Z / p \mathbb Z_{/p} imes \mathbb Z_{/p} Z / p im es Z / p actions.
problem Classifying quotients of S n i m e s S n S^n imes S^n S n im es S n by Z / p i m e s Z / p \mathbb Z_{/p} imes \mathbb Z_{/p} Z / p im es Z / p actions. method Classification via first p p p -localized k k k -invariant, with restrictions on possibilities. result Complete classification of free Z / p i m e s Z / p \mathbb Z_{/p} imes \mathbb Z_{/p} Z / p im es Z / p actions on S 3 i m e s S 3 S^3 imes S^3 S 3 im es S 3 for p > 3 p>3 p > 3 . Let ( X , o ) (X,o) ( X , o ) be a complex normal surface singularity with rational homology sphere link and let X ~ \widetilde{X} X be one of its good resolutions. Fix an effective cycle Z Z Z supported on the exceptional curve and also a possible Chern class l ′ ∈ H 2 ( X ~ , Z ) l'\in H^2(\widetilde{X},\mathbb{Z}) l ′ ∈ H 2 ( X , Z ) . Define E c a l ′ ( Z ) {\rm Eca}^{l'}(Z) Eca l ′ ( Z ) as the space of e…
The paper classifies hypersurfaces in H 2 i m e s H 2 \mathbb{H}^2 imes\mathbb{H}^2 H 2 im es H 2 with constant curvature.
problem Classifying hypersurfaces in H 2 i m e s H 2 \mathbb{H}^2 imes\mathbb{H}^2 H 2 im es H 2 with constant sectional curvature. method Analyzing the geometry of H 2 i m e s H 2 \mathbb{H}^2 imes\mathbb{H}^2 H 2 im es H 2 and constructing specific examples. result Examples of hypersurfaces in H 2 i m e s H 2 \mathbb{H}^2 imes\mathbb{H}^2 H 2 im es H 2 with non-constant product angle function. Study of Sasakian immersions in Heisenberg group and B N i m e s R \mathbb{B}^N imes \mathbb{R} B N im es R , focusing on η η η -Einstein case.
problem Classification of Sasakian immersions in specific spaces.
method Analysis of complete regular Sasakian manifolds in Heisenberg group and B N i m e s R \mathbb{B}^N imes \mathbb{R} B N im es R with standard Sasakian structures. result Complete classification of η η η -Einstein Sasakian immersions. A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.
problem Mapping Fricke-Teichmüller space to character variety of surface representations.
method Bending Fuchsian representations along a fixed measured lamination, proving equivariant symplectic embedding and properness.
result Continuous extension of bending map to Thurston boundary and geometric complexification.
Study CMC hypersurfaces in H 2 i m e s H 2 \mathbb{H}^2 imes\mathbb{H}^2 H 2 im es H 2 with a specific symmetry.
problem Constant mean curvature hypersurfaces with double horocyclic symmetry in H 2 i m e s H 2 \mathbb{H}^2 imes\mathbb{H}^2 H 2 im es H 2 . method Reduction to a single ODE, solving explicitly, classifying solutions.
result Existence and uniqueness of double horocyclic CMC hypersurfaces.
The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.
problem Classifying and constructing differential symmetry breaking operators.
method Utilizing factorization identities and branching laws of generalized Verma modules.
result Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces are classified and constructed.
Complex pinning problem simplified for simple multiloops.
problem Complexity of pinning simple multiloops in fixed surfaces.
method Analysis of pinning problem in simple multiloops.
result Pinning problem in simple multiloops is P when 3 strands or fewer, NP-complete when 20 strands or more.
Uniform bounds on S O ( 2 ) i m e s S O ( 3 ) SO(2) imes SO(3) S O ( 2 ) im es S O ( 3 ) -invariant Ricci solitons on S 4 \mathbb{S}^4 S 4 .
problem Bounding S O ( 2 ) i m e s S O ( 3 ) SO(2) imes SO(3) S O ( 2 ) im es S O ( 3 ) -invariant Ricci solitons on S 4 \mathbb{S}^4 S 4 . method Established uniform constant C \mathcal{C} C for bounded curvature, volume, and injectivity radius. result Strong evidence suggests that only round S O ( 2 ) i m e s S O ( 3 ) SO(2) imes SO(3) S O ( 2 ) im es S O ( 3 ) -invariant Ricci solitons on S 4 \mathbb{S}^4 S 4 exist. New approach to Carrollian geometry using R i m e s \mathbb{R}^ imes R i m es -bundles.
problem Analyzing Carrollian manifolds with degenerate metrics.
method Principal R i m e s \mathbb{R}^ imes R i m es -bundles with degenerate metrics and connections. result Canonical non-degenerate metric derived from principal connection.
Study G 2 G_2 G 2 -instantons on R 4 i m e s S 3 \mathbb{R}^4 imes S^3 R 4 im es S 3 with invariant properties.
problem Existence of G 2 G_2 G 2 -instantons on R 4 i m e s S 3 \mathbb{R}^4 imes S^3 R 4 im es S 3 . method Explicit construction and study of invariant properties.
result Existence of 1-parameter families of G 2 G_2 G 2 -instantons. Study of a G 2 G_2 G 2 -equivariant octonionic operator and its right spectrum.
problem Understanding the spectrum of a G 2 G_2 G 2 -equivariant octonionic operator. method Computed the ordinary real spectrum and analyzed the octonionic right-eigenvalue problem using G 2 G_2 G 2 -decomposition and residual symmetry analysis. result Explicit spectral loci (quartic curve and circle) in each complex slice of the octonionic space.
The paper quantizes hybrid topological-holomorphic field theories on R m i m e s C n \mathbb{R}^m imes \mathbb{C}^n R m im es C n .
problem Quantizing hybrid topological-holomorphic field theories rigorously.
method Constructing perturbative, one-loop quantizations on R m i m e s C n \mathbb{R}^m imes \mathbb{C}^n R m im es C n . result The one-loop obstruction to quantization vanishes when m ≥ 1 m \geq 1 m ≥ 1 . 2D complexes can be almost-embedded in 4D space without self-intersections.
problem Understanding the embedding properties of 2-dimensional complexes in 4-dimensional space.
method Analyzing specific 2-complexes constructed by Freedman-Krushkal-Teichner and showing they can be PL immersed in R 4 \mathbb{R}^4 R 4 without self-intersections. result Many 2-complexes can be PL almost-embedded in R 4 \mathbb{R}^4 R 4 with singularities only as self-intersections of some 2-cells. Alternative proof for links not smoothly slice in S 2 i m e s S 2 S^2 imes S^2 S 2 im es S 2 .
problem Existence of links not smoothly slice in S 2 i m e s S 2 S^2 imes S^2 S 2 im es S 2 . method Alternative proof using Miyazaki and Yasuhara's result.
result Existence of links not smoothly slice in S 2 i m e s S 2 S^2 imes S^2 S 2 im es S 2 . The pinning ideal of multiloops is shown to be NP-complete.
problem Computing the pinning number of multiloops is NP-complete.
method Adapted algorithms from curve intersection and taut loop characterization to prove NP-completeness.
result The pinning number problem is NP-complete for multiloops in the sphere.
Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.
problem Classifying and constructing intertwining differential operators between line and vector bundles over real projective space.
method F-method for classification and construction of intertwining differential operators.
result Generalizes a classical result of Bol for S L ( 2 , R ) SL(2,\mathbb{R}) S L ( 2 , R ) and classifies intertwining operators for S L ( n , R ) SL(n,\mathbb{R}) S L ( n , R ) . Discuss knots in S g i m e s S 1 S_{g} imes S^{1} S g im es S 1 , introducing rotating information and invariants.
problem Understanding knots in S g i m e s S 1 S_{g} imes S^{1} S g im es S 1 and their invariants. method Introduce diagrams, moves, and rotating information to construct invariants.
result Construct invariants using the rotating information of knots in S g i m e s S 1 S_{g} imes S^{1} S g im es S 1 . No Lagrangian Klein bottles found in S 2 i m e s S 2 S^2 imes S^2 S 2 im es S 2 .
problem Existence of Lagrangian Klein bottles in S 2 i m e s S 2 S^2 imes S^2 S 2 im es S 2 . method Luttinger surgery to show non-existence in a specific homology class.
result No Lagrangian Klein bottles in S 2 i m e s S 2 S^2 imes S^2 S 2 im es S 2 . Study of singularities in mean curvature flow with focus on S 3 i m e s R \mathbb{S}^3 imes\mathbb{R} S 3 im es R .
problem Understanding singularities in mean curvature flow.
method Detailed analysis of singularities modeled on S 3 i m e s R \mathbb{S}^3 imes\mathbb{R} S 3 im es R , using normal form transformations and rescaled MCF. result Proves mean convexity and singularity isolation in a small neighborhood, conjectures singularity formation in entire neighborhood.
We classify non-minimal biconservative surfaces with parallel mean curvature vector field in S n × R \mathbb{S}^n\times\mathbb{R} S n × R and H n × R \mathbb{H}^n\times\mathbb{R} H n × R . When these surfaces do not lie in S n \mathbb{S}^n S n or H n \mathbb{H}^n H n and they are not vertical cylinders, we find their explicit (local) equation. We also prove a…
The paper constructs exotic complex projective surfaces using rational blowdowns.
problem Finding exotic complex projective surfaces with specific invariants.
method Rational blowdowns and construction of Kollár--Shepherd-Barron--Alexeev surfaces.
result First exotic surfaces constructed, including minimal and symplectic ones.
Researchers found equations for special surfaces in curved spaces.
problem Identifying biconservative surfaces with non-constant mean curvature.
method Explicit local equations found for surfaces in S 2 i m e s R \mathbb{S}^2 imes\mathbb{R} S 2 im es R and H 2 i m e s R \mathbb{H}^2 imes\mathbb{R} H 2 im es R . result Explicit equations for biconservative surfaces with non-constant mean curvature.
We study 2-dimensional submanifolds of the space L ( H 3 ) {\mathbb{L}}({\mathbb{H}}^3) L ( H 3 ) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in H 3 {\mathbb{H}}^3 H 3 orthogonal to the geodesics of Σ Σ Σ . We prove that the induced metr…
This paper aims to define and study a notion of orientability in the Heisenberg sense ( H \mathbb{H} H -orientability) for the Heisenberg group H n \mathbb{H}^n H n . In particular, we define such notion for H \mathbb{H} H -regular 1 1 1 -codimensional surfaces. Analysing the behaviour of a Möbius Strip in H 1 \mathbb{H}^1 H 1 , we find a 1 1 1 …
We construct non-zero constant mean curvature H surfaces in the product spaces S 2 × R \mathbb{S}^2 \times \mathbb{R} S 2 × R and H 2 × R \mathbb{H}^2\times \mathbb{R} H 2 × R by using suitable conjugate Plateau constructions. The resulting surfaces are complete, have bounded height and are invariant under a discrete group of horizontal translati…
We obtain compact orientable embedded surfaces with constant mean curvature 0 < H < 1 2 0<H<\frac{1}{2} 0 < H < 2 1 and arbitrary genus in S 2 × R \mathbb{S}^2\times\mathbb{R} S 2 × R . These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature 1 2 \frac{1}{2} 2 1 tangent along an equator. This is a particular case of a conjug…
Using the complex parabolic rotations of holomorphic null curves in C 4 {\mathbb{C}}^{4} C 4 , we transform minimal surfaces in Euclidean space R 3 ⊂ R 4 {\mathbb{R}}^{3} \subset {\mathbb{R}}^{4} R 3 ⊂ R 4 to a family of degenerate minimal surfaces in Euclidean space R 4 {\mathbb{R}}^{4} R 4 . Applying our deformation to holomorphic null curves in ${…
Let M 2 \mathbb{M}^{2} M 2 be a complete non compact orientable surface of non negative curvature. We prove in this paper some theorems involving parabolicity of minimal surfaces in M 2 × R \mathbb{M}^{2}\times\mathbb{R} M 2 × R . First, using a characterization of δ δ δ -parabolicity we prove that under additional conditions on M \mathbb{M} M ,…
We prove new local inequality for divisors on surfaces and utilize it to compute α α α -invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type A 1 \mathbb{A}_{1} A 1 , A 2 \mathbb{A}_{2} A 2 , A 3 \mathbb{A}_{3} A 3 , A 4 \mathbb{A}_{4} A 4 , A 5 \mathbb{A}_{5} A 5 or $\mathbb{A}_{6…
Extended Siegel-Jacobi upper half-plane geometry studied with invariant metrics.
problem Characterizing the geometry of the extended Siegel-Jacobi upper half-plane.
method Parameterized using S-coordinates and expressed in terms of invariant metrics.
result Extended Siegel-Jacobi upper half-plane is a reductive, non-symmetric manifold.
The study finds new constant mean curvature surfaces in curved spaces.
problem Finding surfaces with constant mean curvature in curved spaces.
method Analyzing families of surfaces in S 2 i m e s R S^2 imes \mathbb{R} S 2 im es R and H 2 i m e s R H^2 imes \mathbb{R} H 2 im es R . result New families of surfaces with constant mean curvature, including non-equivariant examples.
In this article, we consider compact surfaces Σ Σ Σ having constant mean curvature H H H ( H H H -surfaces) whose boundary Γ = ∂ Σ ⊂ M 0 = M × f { 0 } Γ=\partialΣ\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\} Γ = ∂ Σ ⊂ M 0 = M × f { 0 } is transversal to the slice M 0 \mathbb{M}_0 M 0 of the warped product M × f R \mathbb{M}\times_f\mathbb{R} M × f R , here M \mathbb{M} M denotes a Hadamard surface…
The study classifies constant mean curvature surfaces in curved spaces.
problem Classifying constant mean curvature surfaces in curved spaces.
method Analyzes constant mean curvature isometric immersions into S 2 i m e s R \mathbb{S}^2 imes \mathbb{R} S 2 im es R and H 2 i m e s R \mathbb{H}^2 imes \mathbb{R} H 2 im es R . result Provides new classifications of constant mean curvature surfaces in various curved spaces.
We study area-stationary, or maximal, surfaces in the space L ( H 3 ) {\mathbb L}({\mathbb H}^3) L ( H 3 ) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in L ( H 3 ) {\mathbb L}({\mathbb H}^3) L ( H 3 ) is a maximal surface. We then classify Lagrangian maximal surfa…
Paper investigates reflection principles for zero mean curvature surfaces in isotropic 3-space.
problem Investigating reflection principles for zero mean curvature surfaces in isotropic 3-space.
method Analyzes reflection principles for zero mean curvature surfaces in I 3 \mathbb{I}^3 I 3 . result Shows a reflection principle for isotropic line segments on zero mean curvature surfaces in I 3 \mathbb{I}^3 I 3 . Study defines hyper-dual spheres and ruled surfaces, proving geometric relationships.
problem Understanding geometric properties of hyper-dual spheres and ruled surfaces.
method Defined hyper-dual spheres, developed ruled surfaces, and established geometric relationships.
result Proved isomorphism between hyper-dual sphere and tangent bundle, and geometric interpretation of ruled surfaces.
We simplify and prove a splitting theorem for mapping class groups of connect sums of S 2 i m e s S 1 S^2 imes S^1 S 2 im es S 1 .
problem Understanding the structure of mapping class groups of specific 3-manifolds.
method We prove a splitting theorem for the mapping class group of connect sums of S 2 i m e s S 1 S^2 imes S^1 S 2 im es S 1 , simplifying Laudenbach's original proof. result The mapping class group of connect sums of S 2 i m e s S 1 S^2 imes S^1 S 2 im es S 1 is the semidirect product of Out ( F n ) (F_n) ( F n ) by ( Z / 2 ) n (\mathbb{Z}/2)^n ( Z /2 ) n . In this article we study surfaces in S 3 ( 1 ) × R \mathbb{S}^3(1) \times \mathbb{R} S 3 ( 1 ) × R for which the R \mathbb{R} R -direction makes a constant angle with the normal plane. We give a complete classification for such surfaces with parallel mean curvature vector.
Two classification theorems for Willmore surfaces in S² × S².
problem Classifying Willmore surfaces in S² × S².
method Analytical proofs for minimal and product type surfaces.
result Classification of Willmore surfaces in S² × S².
We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in M 3 ( c ) × R M^3(c)\times\mathbb{R} M 3 ( c ) × R , where M 3 ( c ) M^3(c) M 3 ( c ) is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.
Introduces K \mathbb{K} K -framings for surfaces, generalizing quadratic forms.
problem Generalizing quadratic forms to commutative rings with unit.
method Introduces K \mathbb{K} K -framings and maps based loops to homology classes. result Bijection between K \mathbb{K} K -framings and twisted cocycles for surfaces with positive genus.