arXiv research

Study shows area-minimizing submanifolds are mostly rough, not smooth.

Study shows area-minimizing submanifolds are mostly smooth except for specific types.

Study shows area-minimizing submanifolds are mostly smooth except for geodesics, minimal surfaces, and hypersurfaces.

New submanifolds identified in Spin(7) manifolds with identified deformation spaces.

The paper proves weaker conditions for global smoothings of special Lagrangian submanifolds with conical singularities.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regular…

The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …

Decomposes smooth manifolds into algebraic submanifolds.

Derives Ribaucour coordinates for curves and submanifolds, smoothing curvature line nets.

Constructs smooth isometric extensions for submanifolds.

3-dimensional Harvey Lawson submanifolds were introduced in an earlier paper by Akbulut-Salur, as examples of Lagrangian-type manifolds inside G2 manifold. In this paper, we first show that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with …

Study of filtering and smoothing in submanifolds of Euclidean space.

Smooth functions on manifolds with degenerate singular submanifolds

The paper studies deformations of submanifolds using a new algebraic structure.

Smooth approximation of integral cycles in manifolds.

In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian…

Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.

Proves smoothness of certain Lagrangian submanifolds in complex space.

Bayesian methods estimate regression functions on submanifolds using graph Laplacian eigenbasis.

Study on heat content for submanifolds in sub-Riemannian geometry.

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure …

Extends smoothness results for submanifolds and mean curvature flows with a common boundary.

Solves geometric Cauchy problem for submanifolds with constant rank.

We give some results concerning the smoothness of the image of a real-analytic submanifold in complex space under the action of a finite holomorphic mapping. For instance, if the submanifold is not contained in a proper complex subvariety, we give a necessary and sufficient condition guaranteeing that its image is smoo…

The purpose of this paper is to give an application of the gluing theorem for special Lagrangian submanifolds of a Calabi-Yau 3-fold. We proved a gluing theorem before to smooth a codimension-two singularity of a particular special Lagrangian submanifold. In this paper we will show that this theorem can be applied to m…

The paper extends local calibration pairs to global ones and finds mass-minimizing submanifolds.

Solves geometric submanifold problem for specific distributions.

R.C.McLean showed that the moduli space of nearby submanifolds of a smooth, compact, orientable special Lagrangian submanifold L in a Calabi-Yau manifold X is a smooth manifold and its tangent space at L is identified with the space of harmonic one forms on L. In this paper, we will extend this result from Calabi-Yau m…

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit f…

High codimension submanifolds evolve to convex shapes, leading to smooth limiting flows.

Study proves long-term flow for special submanifolds.

Smoothness of Hamiltonian stationary submanifolds in symplectic manifolds proven.

This is a short, elementary survey article about taut submanifolds. In order to simplify the exposition, we restrict to the case of compact smooth submanifolds of Euclidean or spherical spaces. Some new, partial results concerning taut 4-manifolds are discussed at the end of the text.

Characterizes blowups of Dirac structures on manifolds.

We find all intrinsic measures of $C^{1,1}$ smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding $d$-dimensional spherical Hausdorff measure restricted to the submanifold. The integer $d$ is the degree of the submanifold. These results follow from a different approach to negligi…

We construct examples of $C^\infty$ smooth submanifolds in ${\Bbb C}^n$ and ${\Bbb R}^n$ of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianaly…

The paper proves generic transversality and regularity for minimal submanifolds and area-minimizing currents.

In this paper we complete the study of the normal holonomy groups of complex submanifolds (non nec. complete) of Cn or CPn. We show that irreducible but non transitive normal holonomies are exactly the Hermitian s-representations of [CD09, Table 1] (see Corollary 1.1). For each one of them we construct a non necessaril…

Study of special submanifolds in Page space with constant curvature.

Study on deformations of singular submanifolds in complex geometry.

A very short proof of the following smooth homogeneity theorem of D. Repovs, E. V. Scepin and the author is presented. Let N be a locally compact subset of a smooth manifold M. Assume that for each two points x,y in N there exist their neighborhoods Ux and Uy in M and a diffeomorphism h : Ux \to Uy such that h(x)=y and…

We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-man…

The paper classifies and studies conformal variations of submanifolds.

We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb R}^n$ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable labels too, and to give a new proof of the Galatius--Randal-Williams theorem on the h…

The purpose of this paper is to prove a gluing theorem for a given special Lagrangian submanifold of a Calabi-Yau 3-fold. The proof will be an adaption of the gluing techniques in J-holomorphic curve theory. It is a well known procedure in geometric analysis to construct new solutions to a given nonlinear partial diffe…

In an earlier paper, we proved that, under certain hypotheses, the moduli space of an asymptotically cylindrical special Lagrangian submanifold with fixed boundary of an asymptotically cylindrical Calabi-Yau 3-fold is a smooth manifold. Here we prove the analogous result for an asymptotically cylindrical special Lagran…

This paper proves several natural generalizations of the theorem that for a generic, $C^k$ Riemannian metric on a smooth manifold, there are no closed, embedded, minimal submanifolds with nontrivial jacobi fields.

This is a companion note of [Zhaa] (arXiv:1501.01836) where the extension of local calibration pairs of smooth submanifolds is discussed. Here we emphasize on the case of singular submanifolds. More precisely, we study when a calibration pair around the singular set of a submanifold can extend to a local calibration pa…