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Register a new userCut and singular loci up to codimension 3
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We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension $n-2$ is well known. We go further in this direction by giving a clasification of all points up to a set of Hausdorff dimension $n-3$.
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We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere $S^2$, begin{align*} u_t & = Delta u + | abla u|^2 u quad text{in } Omegatimes(0,T) u &= u_b quad text{on } partial Omegatimes(0,T) u(cdot,0) &= u_0 quad text{in } Omega , end{align*} with $u(x,t): bar Omegatimes [0,T) to S^2$. Here $Omega$ is a bounded, smooth axially symmetric domain in $mathbb{R}^3$. We prove that for any circle $Gamma subset Omega$ with the same axial symmetry, and any sufficiently small $T>0$ there exist initial and boundary conditions such that $u(x,t)$ blows-up exactly at time $T$ and precisely on the curve $Gamma$, in fact $$ | abla u(cdot ,t)|^2 rightharpoonup | abla u_*|^2 + 8pi delta_Gamma text{ as } tto T . $$ for a regular function $u_*(x)$, where $delta_Gamma$ denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a space-codimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng.
In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean metric. One of the main ingredients in our proof is the construction of (unbounded) positive entire solutions for $Delta_{g}phi_lambda=lambda^{2}phi_lambda$, with certain estimates which are uniform for small parameter $lambdain (0,lambda_0)$. In addition, our argument works equally well for semilinear damped wave equations, when the coefficient of the dissipation term is integrable (without sign condition) and space-independent.
We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4-space. We also discuss the case of ribbon k-string links, for $kgeq 3$.
We consider singular foliations of codimension one on 3-manifolds, in the sense defined by A. Haefliger as being Gamma_1-structures. We prove that under the obvious linear embedding condition, they are Gamma_1-homotopic to a regular foliation carried by an open book or a twisted open book. The latter concept is introduced for this aim. Our result holds true in every regularity C^r, r at least 1. In particular, in dimension 3, this gives a very simple proof of Thurstons 1976 regularization theorem without using Mathers homology equivalence.
We consider minimizing harmonic maps $u$ from $Omega subset mathbb{R}^n$ into a closed Riemannian manifold $mathcal{N}$ and prove: (1) an extension to $n geq 4$ of Almgren and Liebs linear law. That is, if the fundamental group of the target manifold $mathcal{N}$ is finite, we have [ mathcal{H}^{n-3}(textrm{sing } u) le C int_{partial Omega} | abla_T u|^{n-1} ,d mathcal{H}^{n-1}; ] (2) an extension of Hardt and Lins stability theorem. Namely, assuming that the target manifold is $mathcal{N}=mathbb{S}^2$ we obtain that the singular set of $u$ is stable under small $W^{1,n-1}$-perturbations of the boundary data. In dimension $n=3$ both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$ with $s in (frac{1}{2},1]$ and $p in [2,infty)$ satisfying $sp geq 2$. We also discuss sharpness.
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