Thickness Formula and C^1-Compactness for C^{1,1} Riemannian Submanifolds

Spin(7)-instantons, Cayley submanifolds, and Fueter sections

We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This recovers an example constructed by Lewis.

A compactness theorem in Finsler geometry

Let (M.F) be a complete Finsler manifold and P be a minimal and compact submanifold of M. Ric_k(x), x in M is a differential invariant that interpolates between the flag curvature and the Ricci curvature. We prove that if on any geodesic c(t) emanating orthogonally from P we have \int_{0}^{\infty}\mathbf{Ric}_{k}(t)>0, then M is compact.

Compactness of the space of minimal hypersurfaces with bounded volume and p-th Jacobi eigenvalue

Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on the p-th Jacobi eigenvalue for p greater or equal than 2 one gains strong convergence to a smooth limit submanifold away from at most p-1 points.

Target-local Gromov compactness

We prove a version of Gromov's compactness theorem for pseudo-holomorphic curves which holds locally in the target symplectic manifold. This result applies to sequences of curves with an unbounded number of free boundary components, and in families of degenerating target manifolds which have unbounded geometry (e.g. no uniform energy threshold). Core elements of the proof regard curves as submanifolds (rather than maps) and then adapt methods from the theory of minimal surfaces.

Restrictions on Submanifolds via Focal Radius Bounds

We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a $C^{1,α}$ compactness result for submanifolds, as well as a "soul-type" structure theorem for manifolds with nonnegative $k^{th}$--intermediate Ricci curvature that have a closed submanifold with dimension $\geq k$ and infinite focal radius. To prove these results, we use the comparison lemma for Jacobi fields from arXiv:1603…

H-compactness of elliptic operators on weighted Riemannian Manifolds

In this paper we study the asymptotic behavior of second-order uniformly elliptic operators on weighted Riemannian manifolds. They naturally emerge when studying spectral properties of the Laplace-Beltrami operator on families of manifolds with rapidly oscillating metrics. We appeal to the notion of H-convergence introduced by Murat and Tartar. In our main result we establish an H-compactness result that applies to elliptic operators with measurable, uniformly elliptic coefficients on weighted Riemannian manifolds. We further discuss the special case of ``locally periodic'' coefficients and st…

Gromov compactness theorem for stable curves

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only continuous and can vary; the curves are only assumed to have fixed ``topological type'', in particular they can be non-closed and the complex structures on them can vary arbitrarily. In connection with this, we study in §2 moduli spaces of nodal curves with boundary and define a natural complex structure for such moduli spaces. We obtain an apriori estimate for pseud…

Compensated Compactness in Banach Spaces and Weak Rigidity of Isometric Immersions of Manifolds

We present a compensated compactness theorem in Banach spaces established recently, whose formulation is originally motivated by the weak rigidity problem for isometric immersions of manifolds with lower regularity. As a corollary, a geometrically intrinsic div-curl lemma for tensor fields on Riemannian manifolds is obtained. Then we show how this intrinsic div-curl lemma can be employed to establish the global weak rigidity of the Gauss-Codazzi-Ricci equations, the Cartan formalism, and the corresponding isometric immersions of Riemannian submanifolds.

Estimates for J-curves as submanifolds

Here we develop some basic analytic tools to study compactness properties of $J$-curves (i.e. pseudo-holomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves, we establish an extrinsic monotonicity principle for non-negative functions $f$ satisfying $Δf\geq -c^2 f$, we show that curves locally parameterized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we establish $ε$-regularity for the squ…

On Manifold Hypothesis: Hypersurface Submanifold Embedding Using Osculating Hyperspheres

Consider a set of $n$ data points in the Euclidean space $\mathbb{R}^d$. This set is called dataset in machine learning and data science. Manifold hypothesis states that the dataset lies on a low-dimensional submanifold with high probability. All dimensionality reduction and manifold learning methods have the assumption of manifold hypothesis. In this paper, we show that the dataset lies on an embedded hypersurface submanifold which is locally $(d-1)$-dimensional. Hence, we show that the manifold hypothesis holds at least for the embedding dimensionality $d-1$. Using an induction in a pyramid …

子流形理论基本定理与超临界低正则性等距浸入

On the fundamental theorem of submanifold theory and isometric immersions with supercritical low regularity

本文提出在超临界低正则性条件下，通过弱莫雷空间理论和补偿紧致性方法，证明了等距浸入的存在性。

论极小曲面与爱因斯坦四维流形之间的平行性

On the Parallels Between Minimal Surfaces and Einstein Four-Manifolds

本文系统梳理极小曲面与爱因斯坦四维流形在变分、拓扑、正则性及分解等方面的平行性，并借助Jensen定理等结果证明某些爱因斯坦四维流形可极小浸入高维球面，如CP²通过Veronese映射嵌入S⁷。

Kahler manifolds and their relatives

Let $M_1$ and $M_2$ be two Kähler manifolds. We call $M_1$ and $M_2$ {\em relatives} if they share a non-trivial Kähler submanifold $S$, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) $h_1: S\to M_1$ and $h_2: S\to M_2$. Moreover, two Kähler manifolds $M_1$ and $M_2$ are said to be {\em weakly relatives} if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds $S_1$ and $S_2$ which admit two Kähler immersions into $M_1$ and $M_2$ respectively. The notions introduced are not equivalent. Our main results in this paper are Theorem 1.…