Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRegularity of soap film-like surfaces spanning graphs in a Riemannian manifold
107
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,alpha}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allards regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a parametric version due to Gromov.
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.
In this paper, we establish that: Suppose a closed Riemannian manifold $(M^n,g_0)$ of dimension $geq 8$ is not locally conformally flat, then the Paneitz-Sobolev constant of $M^n$ has the property that $q(g_0)<q(S^n)$. The analogy of this result was obtained by T. Aubin in 1976 and had been used to solve the Yamabe problem on closed manifolds. As an application, the above result can be used to recover the sequential convergence of the nonlocal Q-curvature flow on closed manifolds recently introduced by Gursky-Malchiodi.
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.
In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the known biharmonic hypersurfaces in a Euclidean sphere, and to prove the non-existence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chens conjecture on biharmonic submanifolds.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
