Study on minimal surfaces with Y-singularities, proving rigidity for Morse index one.
problem Geometric constraints on minimal surfaces with Y-singularities.
method Investigation of surfaces with low Morse index, focusing on Morse index one.
result Partial uniqueness theorem for Y-catenoid with Morse index one.
Proves rigidity in product spaces using index theory.
problem Scalar curvature rigidity in product spaces.
method Fredholm family index theorem.
result Recover corresponding results of Clifford-linear index theory.
New index theory proves Gromov's dihedral conjectures.
problem Comparisons and rigidity of scalar curvatures, mean curvatures, and dihedral angles.
method Developed a new index theory for manifolds with polyhedral boundary.
result Proved Gromov's dihedral extremality and rigidity conjectures.
Study proves rigidity of capillary surfaces in curved 3D spaces.
problem Proving rigidity of capillary surfaces in curved 3D spaces.
method Local rigidity result for infinitesimally rigid capillary surfaces in Riemannian 3-manifolds with mean convex boundary.
result Bounds on genus, boundary components, and area of compact capillary minimal surfaces with low index.
Paper proves rigidity and index of Y-cones in unit ball.
problem Proving rigidity and index of Y-cones in the unit ball.
method Analyzes conformal and minimal immersions of Y-cones meeting the boundary orthogonally.
result Y-cones are the only free boundary minimal surfaces with Morse index 2(n-2).
Rigidity theorem for critical points of Allen-Cahn equation on S³.
problem Rigidity of critical points with low Morse index on S³.
method Analysis of nullity and symmetries of critical points, Frankel-type theorem for nodal sets.
result Critical points with index five are symmetric and vanish on a Clifford torus, realizing the fifth width of the min-max spectrum.
The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian case correspond to one-sided bounds on the sectional curvatures. Starting from 2-d…
Finite index subgroups of relatively hyperbolic groups have equal index.
problem Finite index subgroups of relatively hyperbolic groups have equal index.
method Demonstrating that the number of simplices in a simplicial classifying space grows linearly with index.
result Finite index subgroups of relatively hyperbolic groups have equal index.
New insights into profinite rigidity of Kleinian groups and their subgroups.
problem Characterizing profinite completions of Kleinian groups and their subgroups.
method Analyzing profinite completions and using finite index subgroups to distinguish completions.
result Profinite completions of certain subgroups of finite index in Kleinian groups are not isomorphic.
Proves a quantitative index theorem for positive scalar curvature metrics.
problem Studying conjectures and open questions on positive scalar curvature.
method Quantitative relative index theorem and λ-Lipschitz rigidity theorem. result Positive answers to Gromov's open questions on scalar curvature.
Study finds bifurcation and local rigidity points for solutions to the Yamabe problem on Aloff-Wallach Spaces.
problem Yamabe problem on Aloff-Wallach Spaces
method Constructing 1-parameter families of solutions and examining changes in the Morse index as the parameter varies.
result Identifies bifurcation and local rigidity points for homogeneous solutions to the Yamabe problem.
Counterexample disproves key index computation in Gromov's conjecture paper.
problem Disproving an index computation in Gromov's conjecture paper.
method Constructing a counterexample to an index computation.
result Counterexample disproves the main result of the paper.
We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus g handlebody group into the genus g mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection …
In this paper, we first establish a K-theory version of the equivariant family index theorem for a circle action, then use it to prove several rigidity and vanishing theorems on the equivariant K-theory level.
New rigidity theorems for spin fill-ins with non-negative scalar curvature.
problem Mean curvature rigidity for spin fill-ins with non-negative scalar curvature.
method Two spinorial techniques: extending boundary spinors and comparison using index theory.
result New Witten-type integral inequality for the mass of asymptotically Schwarzschild manifolds.
Scalar curvature rigidity for products of convex hypersurfaces
problem Rigidity of scalar curvature in products of convex hypersurfaces
method Clifford-linear family index theory
result Scalar curvature inequality implies isometry
Proves curvature comparison theorem for manifolds with conical singularities.
problem Comparing scalar mean curvature of manifolds with conical singularities.
method Uses Dirac operator and index theory to prove curvature comparison theorem.
result Proves curvature comparison theorem without knowing the index of the twisted Dirac operator.
New findings confirm parallels to De Giorgi's conjecture for phase transitions in higher dimensions.
problem Understanding phase transitions with bounded index in higher-dimensional spaces.
method Establishing parallels to De Giorgi's conjecture for general solutions of bounded Morse index.
result Finite index solutions to the Allen--Cahn equation in R4 are one-dimensional, and this holds for all 4≤n≤7. Study shows certain infinite-ended groups are not quasi-isometrically rigid.
problem Understanding when infinite-ended groups are quasi-isometrically rigid.
method Combining results on subgroups and hyperbolic groups, adapting Whyte's argument.
result Proves certain infinite-ended groups are not quasi-isometrically rigid.
Paper proves rigidity of manifolds with specific curvature and submanifold properties.
problem Proving rigidity of Riemannian manifolds with certain curvature and submanifold properties.
method Using ancient mean curvature flows to flow out of a minimal submanifold.
result Proves constant sectional curvature of 1 for manifolds with specified properties. We study the rigidity of complete, embedded constant mean curvature surfaces in R^3. Among other things, we prove that when such a surface has finite genus, then intrinsic isometries of the surface extend to isometries of R^3 or its isometry group contains an index two subgroup of isometries that extend.
We consider minimal surfaces M which are complete, embedded and have finite total curvature in R3, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation Δu+f(u)=0inR3. Here f=−W′ with W bistable and balanced, for instance W(u)=41(1−u2)2. We assume that …
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
The paper explores higher fixed point theorems for foliations with applications to rigidity and integrality.
problem Understanding the topological and geometric properties of foliations.
method Applications of higher Lefschetz theorems for foliations, involving Haefliger cohomology.
result The non-triviality of the higher A-hat genus of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions.
The paper proves geometric rigidity using harmonic twisted spinors and scalar curvature comparison.
problem Proving geometric rigidity for closed Riemannian spin manifolds with specific properties.
method Using Gromov's exact-lift two-form method and harmonic spinors to analyze scalar curvature.
result The original metric is Einstein, and the universal cover is real hyperbolic in the positive-spectrum case.
The flip graph and arc complex of a surface are shown to have finite rigidity.
problem Finite rigidity of flip graph and arc complex for surfaces.
method Embedding the flip graph in the arc complex and leveraging finite rigidity of the flip graph.
result Finite rigidity of the flip graph implies finite rigidity of the arc complex.
In this paper, we first establish an S1-equivariant index theorem for Spinc Dirac operators on Z/k manifolds, then combining with the methods developed by Taubes \cite{MR998662} and Liu-Ma-Zhang \cite{MR1870666,MR2016198}, we extend Witten's rigidity theorem to the case of Z/k Spinc manif…
A new method for non-rigid point set registration reduces computational complexity.
problem Efficiently registering non-rigid point sets with large numbers of points.
method Structured Analytic Coherent Point Drift (Analytic-CPD) reformulates CPD for structured analytic mappings.
result Analytic-CPD reduces computational complexity by controlling the deformation model's dimensionality.
Rigidity theorem for scalar curvature on odd-dimensional singular manifolds.
problem Understanding scalar curvature on manifolds with cone-like singularities.
method Analysis of abstract cone operators, spectral flow argument, and twisted Dirac operators.
result Lipschitz rigidity for scalar curvature on Riemannian spin manifolds with cone-like singularities in odd dimensions.
In this paper we show that the only properly immersed self--shrinkers Σ in Rm+1 with Morse index 1 are the hyperplanes through the origin. Moreover, we prove that if Σ is not a hyperplane through the origin then the index jumps and it is at least m+2, with equality if and only if Σ is a cylinder…
Survey of rigidity and gap phenomena in sphere-ball submanifolds.
problem Rigidity and gap phenomena in submanifolds of sphere and ball.
method Comparison of techniques, pinching and gap theorems, Morse index and topology.
result Free boundary condition in ball forces stronger rigidity than in sphere.
Study ancient solutions to free boundary mean curvature flow in convex manifolds.
problem Understanding ancient solutions to free boundary mean curvature flow in convex manifolds.
method Establish rigidity results and construct foliations to describe ancient solutions.
result Ancient solutions to free boundary mean curvature flow are rigid and exhaust all possibilities under certain conditions.
Proves rigidity of stable minimal hypersurfaces in low dimensions.
problem Rigidity of stable minimal hypersurfaces in low dimensions.
method Conformal method inspired by Fischer-Colbrie.
result No stable minimal hypersurfaces in positively curved closed Riemannian manifolds when dimension is 5 or less.
In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than 4π and index at most one…
We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the Morse index of closed f-minimal hypersurfaces isometrically immersed in a general weighted manifold. The technique permits, in particular, to obtain a linear lower bound on the Morse index via the first Betti number for closed f-minimal hyper…
The paper proves rigidity and stability properties of self-shrinking surfaces in 3D space.
problem Rigidity and stability of self-shrinking surfaces in R3. method Analyzing the mean curvature flow and L-index of self-shrinkers. result No stable two-dimensional self-shrinker in R3 exists without properness. New rigidity result for non-orientable manifolds with scalar curvature constraints.
problem Area rigidity for non-orientable manifolds with scalar curvature constraints.
method Developed a technique to extract from a non-vanishing higher index a geometrically useful family of almost D-harmonic sections. result Area rigidity for non-orientable manifolds with scalar curvature constraints.
In this paper we initiate a study of the topological group PPQI(G,H) of pattern-preserving quasi-isometries for G a hyperbolic Poincare duality group and H an infinite quasiconvex subgroup of infinite index in G. Suppose ∂G admits a visual metric d with dimH<dimt+2, where dimH is the Hausd…
In this paper, we consider immersed two-sided minimal hypersurfaces in Rn with finite total curvature. We prove that the sum of the Morse index and the nullity of the Jacobi operator is bounded from below by a linear function of the number of ends and the first Betti number of the hypersurface. When n=4, …
New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.
problem Proving Yau's conjecture for nonlocal minimal surfaces.
method Introducing nonlocal minimal surfaces and applying min-max variational methods.
result Construction of infinitely many nonlocal s-minimal surfaces on closed manifolds. Study on holomorphic curves in 6-sphere with boundary conditions.
problem Characterizing holomorphic curves in nearly-Kähler 6-manifolds with boundary conditions.
method Complex-geometric methods, including second variation formula for area.
result Obtained rigidity results for reflection-invariant holomorphic curves and topological lower bounds for Morse index.
We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary r…
The paper shows that for Coxeter groups, the commensurator of outer automorphisms is rigid.
problem The rigidity of commensurator of outer automorphisms of Coxeter groups.
method Study of the abstract commensurator of the outer automorphism group of a universal Coxeter group.
result For n≥5, the natural map is an isomorphism and every isomorphism between finite index subgroups is conjugation. Let N be at least 4. We prove that every injective homomorphism from the Torelli subgroup into Out(FN) differs from the inclusion by a conjugation in Out(FN). This applies more generally to the following subgroups: every finite-index subgroup of Out(FN) (recovering a theorem of Farb and Handel); every subgro…
In these lecture notes, we combine recent homological methods of Kevin Whyte with older dynamical methods developed by Benson Farb and myself, to obtain a new quasi-isometric rigidity theorem for the mapping class group MCG(S) of a once punctured surface S of genus at least 2: if K is a finitely generated group quasi-i…
We present a connection between minimal surfaces of index one and General Relativity. First, we show that for a certain class of (electro)static systems, each of its unstable horizons is the solution of a one-parameter min-max problem for the area functional, in particular it has index one. We also obtain an inequality…
Paper proves rigidity of minimal disks in 3-balls with non-negative Ricci curvature.
problem Rigidity of free boundary minimal disks in 3-balls with non-negative Ricci curvature.
method Min-max methods and rigidity statements for half-balls with non-negative Ricci curvature.
result Existence and properties of minimal disks with least area in 3-balls.
Study Euler obstruction of 1-forms on determinantal singularities.
problem Understanding the Euler obstruction of 1-forms on determinantal singularities.
method Investigation of connections between local Euler obstruction and PHN index.
result Explicit computations of Euler obstruction for specific singularities.