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Register a new userPartial regularity of harmonic maps from a Riemannian manifold into a Lorentzian manifold
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In this paper, we will study the partial regularity theorem for stationary harmonic maps from a Riemannian manifold into a Lorentzian manifold. For a weakly stationary harmonic map $(u,v)$ from a smooth bounded open domain $OmegasubsetR^m$ to a Lorentzian manifold with Dirichlet boundary condition, we prove that it is smooth outside a closed set whose $(m-2)$-dimension Hausdorff measure is zero. Moreover, if the target manifold $N$ does not admit any harmonic sphere $S^l$, $l=2,...,m-1$, we will show $(u,v)$ is smooth.
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This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $sin(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $sin(0,1)$ and the stationarity/minimality assumption.
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}.
Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-kappa^2$. Using the cone total curvature $TC(Gamma)$ of a graph $Gamma$ which was introduced by Gulliver and Yamada Math. Z. 2006, we prove that the density at any point of a soap film-like surface $Sigma$ spanning a graph $Gamma subset M$ is less than or equal to $frac{1}{2pi}{TC(Gamma) - kappa^{2}area(pmbox{$timeshspace*{-0.178cm}times$}Gamma)}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when $n=3$, this density estimate implies that if begin{eqnarray*} TC(Gamma) < 3.649pi + kappa^2 inf_{pin M} area({pmbox{$timeshspace*{-0.178cm}times$}Gamma}), end{eqnarray*} then the only possible singularities of a piecewise smooth $(mathbf{M},0,delta)$-minimizing set $Sigma$ is the $Y$-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $pi/b$, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.
In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter-symmetric connections; even some of them are not introduced so far. We also find formula for curvature tensor of this new connection.
We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner product, a semi-Riemannian manifold allows the metric tensor to be indefinite on each tangent space, i.e., possessing both positive and negative definite subspaces; differential geometric objects such as geodesics and parallel-transport can be defined on non-degenerate semi-Riemannian manifolds as well, and can be carefully leveraged to adapt Riemannian optimization algorithms to the semi-Riemannian setting. In particular, we discuss the metric independence of manifold optimization algorithms, and illustrate that the weaker but more general semi-Riemannian geometry often suffices for the purpose of optimizing smooth functions on smooth manifolds in practice.
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