Proves a special case of the Gaussian kinematic formula using large sphere limits.
problem Proving a special case of the Gaussian kinematic formula.
method Viewing the GKF as the limit of spherical kinematic formulas for large dimension spheres.
result Proves a special case of the Gaussian kinematic formula.
Minimal surfaces with negative curvature found in large spheres.
problem Existence of minimal surfaces with negative curvature in large dimensional spheres.
method Applied Song's strategy to closed Riemann surfaces with large automorphism groups, resulting in almost hyperbolic minimal surfaces.
result Existence of closed minimal surfaces with negative induced curvature in any sphere of large dimension.
Veronese minimizes normal curvatures to sphere.
problem Bounding normal curvatures of submanifolds.
method Veronese embeddings of projective planes.
result Optimal bound on normal curvatures guarantees sphere.
Existence of harmonic maps from higher-dimensional manifolds to spheres proven.
problem Proving existence of nonconstant harmonic maps from arbitrary manifolds to spheres.
method Using optimal regularity and eigenvalue optimization on manifolds.
result First general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets.
A well known conjecture in the theory of transformation groups states that if p is a prime and (Z/p)^r acts freely on a product of k spheres, then r is less than or equal to k. We prove this assertion if p is large compared to the dimension of the product of spheres. The argument builds on tame homotopy theory for non …
Characterizes kernel interpolation in large dimensions, revealing optimal and sub-optimal regions.
problem Understanding the phase diagram of kernel interpolation in large dimensions.
method Characterization of variance and bias under various source conditions.
result Determined the (s,γ)-phase diagram of large-dimensional kernel interpolation. In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov, and capacity dimension of Buyalo \cite{Buyalo1}) to Nagata-Assouad dimension. This is done by applying two functors on the Lipschitz category of metric spaces: micr…
Rigidity results are obtained for Riemannian d-manifolds with sec⩾1 and spherical rank at least d−2>0. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini--Study metric. This conjecture is verified in all odd dimensions, for …
The paper proves the existence of stable spheres in asymptotically flat 3-manifolds.
problem Existence of stable spheres in asymptotically flat 3-manifolds.
method Lyapunov-Schmidt reduction
result Existence of an asymptotic foliation of (M,g) by stable constant mean curvature spheres. Let G be a finite group acting orthogonally on a pair (S^d,Γ) where Γis a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus g>1 the maximum order of a G-action on a…
We prove that the vector space R^d of any finite dimension d with the standard metric embeds in a bi-Lipschitz way into the group of area-preserving diffeomorphisms G of the two-sphere endowed with the L^p-metric for p>2. Along the way we show that the L^p-metric on the group G is unbounded for p>2 by elementary method…
New energy identity found for biharmonic maps into spheres.
problem Establishing energy identity for biharmonic maps in supercritical dimensions.
method Adapting Lin-Rivière's strategy for sphere-valued maps.
result Energy identity for stationary biharmonic maps into spheres in supercritical dimensions n≥5. We define a large class of integrable nonlinear PDE's, \emph{k-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the soluti…
Study manifolds with positive intermediate Ricci curvature and large symmetry rank.
problem Closed, simply connected manifolds with positive 2nd-intermediate Ricci curvature and large symmetry rank.
method New tools for studying isometric actions on closed manifolds with positive kth-intermediate Ricci curvature, including isotropy rank lemma, symmetry rank bound, and connectedness principle.
result Even dimensional manifolds with large symmetry rank must have trivial odd degree integral cohomology and are either spheres or complex projective spaces.
Study on kernel regression risk in high dimensions using Pinsker bound.
problem Kernel regression risk in high-dimensional inner product spaces.
method Investigation of Pinsker bound for kernel regression on sphere Sd with sample size n=αdγ(1+od(1)). result Exact minimax risk and Pinsker constant identified for kernel regression.
It is a consequence of the classical Jordan bound for finite subgroups of linear groups that in each dimension n there are only finitely many finite simple groups which admit a faithful, linear action on the n-sphere. In the present paper we prove an analogue for smooth actions on arbitrary homology n-spheres: in each …
The paper proves the regularity of cohomogeneity two problems and constructs minimal hypersurfaces on spheres.
problem Cohomogeneity two equivariant isotopy minimization problems and minimal hypersurfaces with large Betti numbers.
method Developed cohomogeneity two equivariant min-max theory for minimal hypersurfaces.
result Constructs minimal hypersurfaces on spheres with large Betti numbers and specific symmetries.
The study finds conditions for area-minimizing cones over submanifolds.
problem Conditions for area-minimizing cones over submanifolds.
method General configuration results for area-minimizing cones.
result Cone over the minimal product of submanifolds and spheres are area-minimizing.
New cube complexes disprove Kalai's conjecture about sphere facets.
problem Disproving Kalai's conjecture about the number of facets of cubical spheres.
method Constructing cube complexes homeomorphic to the d-sphere with n vertices and Ω(n^(5/4)) facets.
result The conjecture is disproved for all d≥3 and n sufficiently large.
In a recent article the first three authors proved that in dimension 4m+1 all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension 4m−1,m≥2 admit Sasakian-…
Study proves mean curvature flows on spheres in higher dimensions.
problem Existence of mean curvature flows on spheres.
method Generalized previous results to higher dimensions, proving existence of flows.
result Existence of infinitely many eternal weak mean curvature flows in Sn+1 connecting specific hypersurfaces. Existence and instability of biharmonic maps from balls to spheres.
problem Existence and stability of biharmonic maps between balls and spheres.
method Existence proof and instability analysis using bienergy.
result Existence of two proper biharmonic maps and instability in low dimensions.
The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.
problem Understanding the number of terms in the connected sums of sphere products associated with dual-neighborly polytopes.
method Combinatorial operations and formulas for the number of terms in the connected sums of sphere products.
result Formulas for the number of terms in the connected sums of sphere products associated with dual-neighborly polytopes.
Study of quasilocal energy in higher dimensions, focusing on small sphere limits.
problem Understanding quasilocal energy in higher dimensions and its small sphere limits.
method Generalized quasilocal energy definitions, evaluated along lightcone cuts, and compared with known energies.
result The small sphere limits of quasilocal energy in higher dimensions are not proportional to the Bel-Robinson superenergy, challenging its role as gravitational energy.
Arboricity of manifolds is explored, with specific results for 2D surfaces.
problem Understanding the arboricity of different types of manifolds.
method Analyzing discrete 2-spheres, other 2D surfaces, and d-manifolds of higher dimensions.
result Arboricity of 2D surfaces is 3 or 4, and for higher dimensions, it can be arbitrarily large.
The study classifies transverse spheres in flag manifolds and finds new examples.
problem Classifying transverse spheres in flag manifolds.
method Using topological K-theory and constructions of transverse spheres.
result Classification of transverse spheres in various flag manifolds.
This paper is devoted to dualization of dimension-theoretical results from the small scale to the large scale. So far there are two approaches for such dualization: one consisting of creating analogs of small scale concepts and the other amounting to the covering dimension of the Higson corona ν(X) of X. The first …
We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.
The paper solves the existence problem of sphere packings in higher dimensions.
problem Existence of crystallographic sphere packings in certain higher dimensions.
method Geometric doubling procedure and computations with Lorentzian quadratic forms.
result Solves the existence problem of crystallographic sphere packings in higher dimensions.
New method for triharmonic maps to spheres in various dimensions.
problem Creating triharmonic maps to spheres in different dimensions.
method Construction method based on eigenmaps and suitable deformations.
result Existence of triharmonic maps from Rm∖{0} into spheres. In this article we prove that, for an oriented PL n-manifold M with m boundary components and d0∈N, there exist mutually disjoint closed Euclidean balls and a K-quasiregular mapping M→Sn∖int(B1∪⋯∪Bm) of degree at least d0. The result is …
Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.
problem Rigidity of minimal Legendrian submanifolds in unit Euclidean spheres.
method Using Lu's inequality and eigenvalues of fundamental matrices to establish pinching theorems.
result Optimal pinching theorem and rigidity theorem for submanifolds of all dimensions.
We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale …
In low dimensions, minimizers for the second conformal eigenvalue do not exist near the round sphere.
problem Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions.
method Analysis of conformal classes and renormalized volume in dimensions 3 to 10.
result Existence of minimizers is proven not to hold for metrics sufficiently close to the round metric on the sphere in dimensions 3 to 10.
We prove the existence of exotic but homotopically trivial contact structures on spheres of dimension 8k-1. Together with previous results of Eliashberg and the second author this establishes the existence of such structures on all odd-dimensional spheres (of dimension at least 3).
For odd-dimensional spheres, there's always a second short geodesic.
problem Finding the second shortest closed geodesic on odd-dimensional spheres.
method Analyzing non-reversible Finsler metrics on spheres of odd dimension.
result There is a second closed geodesic with Morse index ≤ 4(m+2)(m-1)+2.
The paper constructs many knotted and linked objects in higher dimensions.
problem Understanding knotted and linked objects in higher dimensions.
method Using barbell diffeomorphisms to construct examples.
result Infinitely many knotted and linked objects in 4 and 5 dimensions.
Numerical discovery matches eta invariant on Berger spheres with conformal anomaly on round spheres.
problem Matching eta invariants on Berger spheres and round spheres.
method Numerical discovery and analytical expression derivation.
result Eta invariant on Berger spheres matches conformal anomaly on round spheres.
The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.
problem Understanding the homotopy types of diffeomorphism groups and sphere embeddings.
method Cerf's upgraded proof, scanning maps, canceling handles, Embedding Calculus.
result The monoid of Schoenflies spheres forms a group under connect-sum.
The paper studies how submanifolds of a sphere evolve over time.
problem Evolution of pinched submanifolds in the sphere.
method High codimension mean curvature flow with pinching conditions.
result Convergence to a round point or totally geodesic sphere under pinching conditions.
The paper finds infinite families of exotic spheres with free actions.
problem Detecting smooth free S1 and S3 actions on exotic spheres. method Topological modular forms.
result Infinite families of very exotic spheres with free actions.
We prove the existence of Sasakian-Einstein metrics on infinitely many rational homology spheres in all odd dimensions greater than 3. In dimension 5 we obain somewhat sharper results. There are examples where the number of effective parameters in the Einstein metric grows exponentially with dimension.
New solutions found for a complex boundary problem.
problem One-phase free boundary problem in geometry.
method New connection with minimal surfaces.
result Disproved a conjecture and constructed homogeneous solutions.
Minimal sphere dimension for equivariant embedding of circles.
problem Embedding a bouquet of circles into a sphere.
method Finding the minimal dimension of the sphere for equivariant embedding.
result The minimal dimension is 2g−1. Study shows curvature rigidity of specific metric types.
problem Curvature rigidity of specific metric types.
method Spin geometry based arguments.
result Scalar curvature rigidity of specific metric types.
New Zoll families of minimal spheres found in spheres and projective spaces.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
In 1987, Kalai proved that stacked spheres of dimension d≥3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d=2. In this article, we give a characterisation of stacked 2-spheres using what we call the {\em separatio…
Let f be a smooth map between unit spheres of possibly different dimensions. We prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions. A corollary is that any area-decreasing map between unit spheres (of possibly different dimensions) is homotopic to a constant…