arXiv research

Stable planes are locally isomorphic to classical projective planes.

Three types of Einstein metrics are disqualified as potential local maxima.

We consider the notion of stable isomorphism of bundle gerbes. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H^3(M, Z). Stable isomorphism sheds light on the local theory of bundle gerbes and enables us to develop a classifying theory …

Locally stable maps $S^3\to\mathbb{R}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $σ(f)$ of its framed singularity link, the generalized normal degree $ν(f)$, and the algebraic number of cusps $κ(f)$ of any extensi…

We will consider locally conformally balanced manifolds. We prove that a locally conformally balanced condition is not stable under a small deformation. We prove that locally conformally balanced condition is stable under any proper modification. We prove that symmetric products of the Kodaira surface can be resolve to…

The paper explores stable surfaces in Einstein-Maxwell theory, proving mass bounds and nonexistence results.

The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.

Exponential growth of stable subgroups in Morse geodesics.

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and …

Study virtual fundamental classes of derived manifolds, proving invariant vanishes.

Local minimality proven for stable free-boundary minimal hypersurfaces.

The paper proves volume comparison theorems for metrics near stable Einstein and Ricci flat manifolds.

The paper establishes a correspondence between Higgs torsors and connections on curves.

The paper studies deformations of singular minimal hypersurfaces in dimensions 7 and above.

Study proves area estimates for stable capillary hypersurfaces with nonpositive Yamabe invariant.

In this paper we show how the existence of a certain stable cylinder determines (locally) the ambient manifold where it is immersed. This cylinder has to verify a {\it bifurcation phenomena}, we make this explicit in the introduction. In particular, the existence of such a stable cylinder implies that the ambient manif…

We establish some a priori geometric relations on stable minimal surfaces lying inside three-manifolds with scalar curvature uniformly bounded below. The relations are based on a slight generalization of a formula due to Castillon. We apply it to prove non-local rigidity results in the particular sense that they expres…

We show that moduli spaces of stable maps admits virtual orbifold structure. The symplectic version of virtual localization formula is obtained.

We present an approach to Gromov-Witten invariants that works on arbitrary (closed) symplectic manifolds. We avoid genericity arguments and take into account singular curves in the very formulation. The method is by first endowing mapping spaces from (prestable) algebraic curves into the symplectic manifold with the st…

Graph products inherit Morse local-to-global property from their components.

The gradient flow of the Yang-Mills action acts pointwise on closed loops of gauge fields. We construct a topologically nontrivial loop of SU(2) gauge fields on S4 that is locally stable under the flow. The stable loop is written explicitly as a path between two gauge fields equivalent under a topologically nontrivial …

Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.

Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete…

Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the re…

New method calibrates LV surfaces for exotic derivatives with smoother, more stable Greeks.

Characterizes values of slice-torus invariants related to knot genus.

Local gluing connects flow lines in finite time intervals.

We prove that a stable minimal hypersurface of an open ball having a singular set of locally finite codimension 2 Hausdorff measure which is weakly close to a multiplicity 2 hyperplane is a 2-valued C^{1, alpha} graph in the interior. Applications including a compactness theorem for a class of immersed stable minimal h…

New method stabilizes quantum ergodicity for mixed quantization and partial hyperbolicity.

Flat minimal hypersurfaces found in wedge-shaped domains.

New definition of stable $(r+1)$-th capillary hypersurfaces proposed.

We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.

The study examines stable regions in weighted manifolds with boundary properties.

Unified approach classifies stable and minimal elastic curves.

Groups of importance in group theory have flexible stability properties.

We introduce a new method of calculating intersections on \bar{M}_{g,n}, using localization of equivariant cohomology. As an application, we give a proof of Mirzakhani's recursion relation for calculating intersections of mixed psi and kappa_1 classes.

Adaptive importance sampling for estimating point process statistics.

We prove that a strictly stable minimal $C^2_h$ intrinsic graph G is locally area-minimizing, i.e. given any $C^1_h$ graph $S$ with the same boundary, $\text{Area}(G)<\text{Area}(S)$ unless $G=S$. As a consequence we show the existence and the uniqueness of $C^\infty$ minimal graphs with prescribed small boundary datum…

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

Topology of non-orientable spaces without boundary is studied.

Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.

Generalized complex (GC) geometry interpolates between ordinary symplectic and complex geometry. Stable generalized complex manifolds (first introduced by Cavalcanti, Gualtieri in 2015) carry a Poisson structure which is generically symplectic, but degenerates on a (real) codimension-2 submanifold. Up to gauge equivale…

Gradient descent converges linearly in finite-width networks with positive NTK and compatible conditions.

For $L \hookrightarrow X$ a Lagrangian embedding associated with a real homogeneous space, we construct the moduli space of stable holomorphic discs mapping to $(X,L)$ as an orbifold with corners equipped with a group action. Some essential constructions involving orbifolds with corners are also discussed, including th…

The paper studies HYM connections on stable vector bundles over Kähler manifolds.

For a generic embedding of a smooth closed surface $M$ into $\mathbb R^4$, the subset of $\mathbb R^4$ which is the affine $λ$-equidistant of $M$ appears as the discriminant set of a stable mapping $M \times M \to \mathbb R^4$, hence their stable singularities are $A_k, \, k=2, 3, 4,$ and $C_{2,2}^{\pm}$. In this paper…

We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.