We introduce techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and blow ups) into more effective control. In the present paper, we focus on proving regularity theorems for stationary and minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on their Minkowski r-content, or equivalently, on the volumes of their r-tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps f : M->N between riemannian manifolds include a priori bounds in W^{1,p}\cap W^{2,p} for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control is actually provides much stronger information than follows from a bound on the L^p norm of derivatives. Namely, we obtain L^p bounds for the inverse of the regularity scale r_f(x)=max{r:sup_{B_r(x)}(r|\nabla f|+r^2|\nabla^2 f|)\leq 1}. Applications to minimal hypersufaces include a priori L^p bounds for the second fundamental form A for all p < 7. Previously known bounds were for p < 2+ε(n). Again, the full theorem is much stronger and yields Lp bounds for the corresponding regularity scale r_A(x)=max{r:sup_{B_r(x)}rA\leq 1}. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key point in all of these arguments is to establish the relevant quantitative differentiation theorem.
In this paper we prove quantitative regularity results for stationary and minimizing extrinsic biharmonic maps. As an application, we determine sharp, dimension independent $L^p$ bounds for $\nabla^k f$ that do not require a small energy hypothesis. In particular, every minimizing biharmonic map is in $W^{4,p}$ for all $1\le p<5/4$. Further, for minimizing biharmonic maps from $Ω\subset \mathbb{R}^5$, we determine a uniform bound on the number of singular points in a compact set. Finally, using dimension reduction arguments, we extend these results to minimizing and stationary biharmonic maps …
The aim of this note is to extend the results in arXiv:1504.02043 to the case of approximate harmonic maps. More precisely, we will proved that the singular strata $S^k(u)$ of an approximate harmonic map are k-rectifiable, and we will show effect bounds on the quantitative strata. In the process we will simplify many of the arguments from arXiv:1504.02043, and in particular we produce a new main covering lemmas which vastly simplifies the older argument.
We prove partial regularity of stationary solutions and minimizers $u$ from a set $Ω\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_ΩF(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an $ε$-regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable.
We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our methods also lead to the full interior $ε$-regularity and smooth estimates for weakly Dirac-harmonic maps in all dimensions.
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
We consider Type I Ricci flows and obtain integral estimates for the curvature tensor valid up to, and including, the singular time. Our estimates partially extend to higher dimensions a curvature estimate recently shown to hold in dimension three by Kleiner and Lott. To do this we adapt the technique of quantitative stratification, introduced by Cheeger and Naber, to this setting.
We introduce a heat flow associated to half-harmonic maps, which have been introduced by Da Lio and Rivière. Those maps exhibit integrability by compensation in one space dimension and are related to harmonic maps with free boundary. We consider a new flow associated to these harmonic maps with free boundary which is actually motivated by a rather unusual heat flow for half-harmonic maps. We construct then weak solutions and prove their partial regularity in space and time via a Ginzburg-Landau approximation. The present paper complements the study initiated by Struwe and Chen-Lin.
We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald.
We investigate the logarithmic and power-type convexity of the length of the level curves for $a$-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the $p$-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the $W^{2,2}$ regularity. The results are complemented by a number of estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes r…
In a recent paper the author introduced a new method based on viscosity techniques for producing minimal surfaces by minmax arguments. The present work corresponds to the regularity part of the method. Precisely we establish that any weakly conformal $W^{1,2}$ map from a riemann surface $S$ into a closed oriented sub-manifold $N^n$ of an euclidian space ${\mathbb R}^m$ realizes a stationary varifold if and only if it is a smooth conformal harmonic map form $S$ into $N^n$.
In my previaou paper of K. Horihata, we have proposed a Ginzburg-Landau system with a time-dependent parameter and then passing to the limit we have constructed a harmonic heat flow into spheres. Thanks to this scheme, we establish a few energy inequalities of our flow: (i) monotonical inequalities and (ii) a reverse Poincare inequality at any boundary point. These inequalities (i) and (ii) derive the smaller estimates on the set on noncontinuouspoints for pur flow contrast to the former results. We refer to them by Y. Chen and Y. Chen, J. Li, F. H. Lin. Next we introduce two sufficient condit…
This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harm…
Energy minimizing harmonic maps between manifolds are known to be smooth outside a rectifiable set of codimension $3$, called the singular set. The possibility that this set is not a manifold, but has arbitrarily many small gaps in it, is not excluded in general. Here we prove that some part of the singular set - characterized by topological and analytic properties of tangent maps - is a topological manifold. In the special case of maps into the sphere ${\mathbb S}^2$, we conclude that the whole top-dimensional part of the singular set is a manifold - this generalizes a similar result in two-d…
In this article, we study the regularity of minimizing and stationary $p$-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set $S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous at } x\}$, as opposed to the weaker and non quantitative Hausdorff dimension bounds currently available in literature for generic $p$. The main technique used in this paper is the quantitative stratification, which is based on the study of the approximate symmetries of the tangent maps of $f$. In this article, we generalize the study carried out in \cite{Ch…
In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in \cite{lin97}. The proof extends the argument of Huang-Wang \cite {hua-w10}.
We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by Cheeger-Naber.
We determine regularity results for energy minimizing maps from an $n$-dimensional Riemannian polyhedral complex $X$ into a CAT(1) space. Provided that the metric on $X$ is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $(n-2)$-skeleton, we improve the regularity to locally Lipschitz. Finally, for points $x \in X^{(k)}$ with $k \leq n-2$, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of $x \in X$.
In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is $k^{th}$-stratum of the singular set of $f$, then it is well known that $\dim S^k\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable. In the case of minimizing harmonic maps we go further, and prove that the singular set $S(f)$…
We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconformal maps are smooth. In particular, we prove that contact manifolds support the suitable regularity. The main new technical tools are a sub-Riemannian version of p-harmonic coordinates and a technique of propagation of regularity from horizontal layers.
$α$-Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $α$-harmonic maps that were introduced by Sacks-Uhlenbeck to attack the existence problem for harmonic maps from surfaces. For $α>1$, the latter are known to satisfy a Palais-Smale condtion, and so, the technique of Sacks-Uhlenbeck consists in constructing $α$-harmonic maps for $α>1$ and then letting $α\to 1$. The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $α$-Dirac-harmonic maps when …
We study conformal harmonic coordinates on Riemannian manifolds. These are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show their existence under general conditions. We find that conformal harmonic coordinates are a close conformal analogue of harmonic coordinates. We prove up to boundary regularity results for conformal mappings. We show that Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors become elliptic operators in conformal harmonic coordinates if one normalizes the determinant of the metric. We give a corresponding elliptic regular…
We prove that the singular set of an energy-minimizing map from Euclidean space into an $F$-connected complex is $(m-2)$-rectifiable. This strengthens the regularity result of Gromov and Schoen.
We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^α$ metric tensors is $C^{1+α}$ regular, and that a manifold with $W^{1,n} \cap C^α$ metric tensor and with vanishing Weyl tensor is locally conformally flat if $n \geq 4$. The results extend the works [LS14, LS15] from the case of $C^{1+α}$ metrics to the Hölder continuous case. In…
