arXiv research

Maps between positively curved manifolds with non-increasing area are rigid.

New rigidity result for non-orientable manifolds with scalar curvature constraints.

We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors. We show that k'\ge k everywhere on M implies k'=k. Under an additional condition on th…

In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in C^{\infty} sense as time goes into infinity.

Study a flow preserving area of plane curves, ending in a circle.

Shows large unknotting number for simple knots.

New rigidity result for maps between curved spaces.

Scalar curvature rigidity for products of convex hypersurfaces

The paper proves rigidity for certain product spaces and bounds for band widths.

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), …

We show that if G is a discrete subgroup of the group of the isometries of the hyperbolic k-space H^k, and if R is a representation of G into the group of the isometries of H^n, then any R-equivariant map F from H^k to H^n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of thi…

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (…

BiLipschitz mappings can be extended to preserve area.

Let $B_1$ be the unit open disk in $\Real^2$ and $M$ be a closed Riemannian manifold. In this note, we first prove the uniqueness for weak solutions of the harmonic map heat flow in $H^1([0,T]\times B_1,M)$ whose energy is non-increasing in time, given initial data $u_0\in H^1(B_1,M)$ and boundary data $γ=u_0|_{\partia…

Given a compact Alexadrov $n$-space $Z$ with curvature curv $\ge κ$, and let $f: Z\to X$ be a distance non-increasing onto map to another Alexandrov $n$-space with curv $\ge κ$. The relative volume rigidity conjecture says that if $X$ achieves the relative maximal volume i.e. $vol(Z)=vol(X)$, then $X$ is isometric to $…

A quaternionic contact (qc) heat equation and the corresponding qc energy functional are introduced. It is shown that the qc energy functional is monotone non-increasing along the qc heat equation on a compact qc manifold provided certain positivity conditions are satisfied.

We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF_4 by enumerating the connected components of the moduli space of real 2-elementary K3 surfaces of type (S,θ)=((3,1,1), -id). We also list up all the ca…

Let $M^n$ be a complete, non-compact and $C^\infty$-smooth Riemannian manifold with nonnegative sectional curvature. Suppose $\Cal S$ is a soul of $M^n$. Then any distance non-increasing retraction $Ψ: M^n \to \Cal S$ must give rise to a $C^\infty$-smooth Riemannian submersion.

The study proves that sets with constant nonlocal curvature are composed of equal balls under certain conditions.

The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.

Gradient flow method solves isoperimetric inequality for maps.

We consider the mean curvature flow of the graph of a smooth map $f:\mathbb{R}^2\to\mathbb{R}^2$ between two-dimensional Euclidean spaces. If $f$ satisfies an area-decreasing property, the solution exists for all times and the evolving submanifold stays the graph of an area-decreasing map $f_t$. Further, we prove unifo…

The study finds a continuous map achieving minmax area under Legendrian constraints.

The paper extends the avoidance principle for mean curvature flows, proving new intersection dimension monotonicity results.

We consider a closed manifold M with a Riemannian metric g(t) evolving in direction -2S(t) where S(t) is a symmetric two-tensor on (M,g(t)). We prove that if S satisfies a certain tensor inequality, then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latter be…

Characterizes area-minimizing maps for surfaces of genus ≥ 2.

An ODE variational calculation shows that an image principle curvature ratio factor can raise the lower bound, 2(Image Area), on energy of a harmonic map of a surface into Rn. In certain situations, including all radially symmetry harmonic maps, equality is achieved.

Study of area minimizing surfaces in homotopy classes of maps.

This paper sets a lower bound for the Gauss map area of surfaces in S^3.

Let $f:M\to N$ be a smooth area decreasing map between two Riemannian manifolds $(M,\gm)$ and $(N,\gn)$. Under weak and natural assumptions on the curvatures of $(M,\gm)$ and $(N,\gn)$, we prove that the mean curvature flow provides a smooth homotopy of $f$ to a constant map.

Study uniformly differentiable graphs in Carnot groups, proving area formulas.

We discuss a special class of solutions to the minimal surface system. These are vector-valued functions that "decrease area" and are natural generalization of scalar functions. After defining area-decreasing maps, we show several classical results for the minimal surface equation can be generalized. We also conjecture…

Lipschitz maps on metric surfaces are rigid if they preserve area.

In this paper, we study the evolution of L2 p-forms under Ricci flow with bounded curvature on a complete non-compact or a compact Riemannian manifold. We show that under curvature pinching conditions on such a manifold, the L2 norm of a smooth p-form is non-increasing along the Ricci flow. The L^{\infty} norm is showe…

New algorithm speeds up online mapping of unknown terrains.

Solves Dirichlet problem for harmonic maps to give geodesic insights.

Study of loss functions for learning to defer, proving consistency.

We prove that higher moment maps on area measures of a euclidean vector space are injective, while the kernel of the centroid map equals the image of the first variation map. Based on this, we introduce the space of smooth dual area measures on a finite-dimensional euclidean vector space and prove that it admits a natu…

The coarea formula is proven for Heisenberg group maps, addressing open questions.

In this note, we obtain the asymptotic estimate for the time derivative of the $Φ$-entropy in terms of the lower bound on the Bakry-Emery $Γ_2$ curvature. In the cases of Hyperbolic space and Heisenberg group, we show that the time derivative of the $Φ$-entropy is non-increasing, and we also get sharp asymptotic bound …

Minimal surfaces in 4D space are stable if their Gauss map spherical area is less than 2π.

Characterizes harmonic morphisms preserving minimal submanifolds and finds novel area-minimising hypercones.

We identify a class of over-parameterized deep neural networks with standard activation functions and cross-entropy loss which provably have no bad local valley, in the sense that from any point in parameter space there exists a continuous path on which the cross-entropy loss is non-increasing and gets arbitrarily clos…

We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove…

In this study we investigate the potential for using synthetic aperture radar (SAR) data to provide high resolution defoliation and regrowth mapping of trees in the tundra-forest ecotone. Using aerial photographs, four areas with live forest and four areas with dead trees were identified. Quad-polarimetric SAR data fro…

Maps on foliated manifolds decrease area and scalar curvature is negative.

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface…