arXiv research

The abstract extends deformation and tangent groupoid constructions to infinite-dimensional manifolds.

Developing a non-symmetric strainer theory for spaces with non-negative curvature beyond Alexandrov geometry.

In this note we show that for any proper action of a Banach--Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps $\g \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $…

The article presents a description of geometry of Banach structures forming mathematical base of markets arbitrage absence type phenomena. In this connection the role of reflexive subspaces (replacing classically considered finite-dimensional subspaces) and plasterable cones is uncovered.

Unique cylindrical tangent cone for Simons' hypersurface found.

Analytic sets with unique infinite tangent cone are algebraic.

We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent …

Uniqueness proven for stable hypersurface tangent cones.

Uniqueness proven for cylindrical tangent cones in high dimensions.

New Calabi-Yau metrics found on complex symmetric spaces.

Researchers prove uniqueness of certain cylindrical tangent cones for special Lagrangians.

Study on measure-valued CARMA processes in Banach spaces.

A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…

Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to …

New examples of Ricci limit spaces with mixed tangent cones.

Develops a new method to study algebraic tangent cones of sheaves using valuations.

Study proves existence of precotangent bundles for Grassmannians.

Kähler-Ricci flows' tangent cones are algebraic varieties.

Paper proves unique tangent maps for complex maps into algebraic varieties.

Study shows unique tangent cones for area-minimizing currents at boundary points.

Study of tangent cones at infinity for algebraic sets.

Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.

New special Lagrangian submanifolds with cylindrical tangent cones are constructed.

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities.

New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.

Generalizes Connes's tangent groupoid for sub-Riemannian geometry.

Study extends JB-algebra structure group results to infinite dimensions.

Bounded symmetric domains are biholomorphic to tube domains over Finsler symmetric cones.

Banach fibrations and Nijenhuis operators studied for vanishing torsion.

A Banach symmetric space in the sense of O. Loos is a smooth Banach manifold $M$ endowed with a multiplication map $μ\colon M \times M \to M$ such that each left multiplication map $μ_x := μ(x,\cdot)$ (with $x \in M$) is an involutive automorphism of $(M,μ)$ with the isolated fixed point $x$. We show that morphisms of …

The paper studies symplectic forms on projective limits of Banach bundles and their Darboux Theorem.

It is shown by Colding and Minicozzi the uniqueness of the tangent cone at infinity of Ricci-flat manifolds with Euclidean volume growth which has at least one tangent cone at infinity with a smooth cross section. In this article we raise an example of the Ricci-flat manifold implying that the assumption for the volume…

Let $X$ be a (real or complex) Banach space, and $\mathcal{I}(X)$ be the set of all (non-zero and non-identity) idempotents; i.e., bounded linear operators on $X$ whose squares equal themselves. We show that the Banach submanifold $\mathcal{I}(X)$ of $\mathcal{L}(X)$ is a locally trivial analytic affine-Banach bundle o…

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connecti…

The study shows that certain metrics on spheres prevent stable tangent cones for area-minimizing boundaries.

For a compact, smooth C^r orbifold (without boundary), we show that the topological structure of the orbifold diffeomorphism group is a Banach manifold for finite r \ge 1 and a Frechet manifold if r=infty. In each case, the local model is the separable Banach (Frechet) space of C^r (C^infty, resp.) orbisections of the …

Consider a limit space $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α^n$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of $Y$ at a point $p\in Y$ are known to be metric cones $C(X)$, however they need not be unique. Let $\barΩ_{Y,p}\subseteq\cM_{GH}$ be the close…

In this paper we give a complete algebro-geometric characterization of analytic tangent cones of admissible Hermitian-Yang-Mills connections over any reflexive sheaves.

Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.

We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].

The paper constructs 4-manifolds with nonnegative Ricci curvature and specific asymptotic cones.

Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in X$, the tangent cone of…

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform r…

The study examines singularities in flows with curvature bounds and identifies unique tangent flows.

Unique tangent cones found for Kahler-Einstein metrics on singular varieties.

Study on Kähler-Einstein metrics with polynomial convergence rates.

Study proves uniqueness of Yang-Mills field tangent cones in arbitrary dimensions.

Study on singularities in area-minimizing currents, proving unique tangent cones and rectifiability.