Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTranslating solitons over Cartan-Hadamard manifolds
81
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions.
rate research
Read More
We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance to a fixed point in M. We are, in particular, interested in finding optimal (or close to optimal) curvature upper bounds.
The generalized Cartan-Hadamard conjecture says that if $Omega$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K le kappa le 0$, then the boundary of $Omega$ has the least possible boundary volume when $Omega$ is a round $n$-ball with constant curvature $K=kappa$. The case $n=2$ and $kappa=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $kappa=0$, and a special case of the conjecture for $kappa textless{} 0$ and a version for $kappa textgreater{} 0$. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Crokes proof for $n=4$ and $kappa=0$. The generalization to $n=4$ and $kappa e 0$ is a new result. As Croke implicitly did, we relax the curvature condition $K le kappa$ to a weaker candle condition $Candle(kappa)$ or $LCD(kappa)$.We also find counterexamples to a naive version of the Cartan-Hadamard conjecture: For every $varepsilon textgreater{} 0$, there is a Riemannian 3-ball $Omega$ with $(1-varepsilon)$-pinched negative curvature, and with boundary volume bounded by a function of $varepsilon$ and with arbitrarily large volume.We begin with a pointwise isoperimetric problem called the problem of the Little Prince. Its proof becomes part of the more general method.
We study the fourth order Schrodinger equation with mixed dispersion on an $N$-dimensional Cartan-Hadamard manifold. At first, we focus on the case of the hyperbolic space. Using the fact that there exists a Fourier transform on this space, we prove the existence of a global solution to our equation as well as scattering for small initial data. Next, we obtain weighted Strichartz estimates for radial solutions on a large class of rotationally symmetric manifolds by adapting the method of Banica and Duyckaerts (Dyn. Partial Differ. Equ., 07). Finally, we give a blow-up result for a rotationally symmetric manifold relying on a localized virial argument.
We state and prove a Chern-Osserman-type inequality in terms of the volume growth for complete surfaces with controlled mean curvature properly immersed in a Cartan-Hadamard manifold $N$ with sectional curvatures bounded from above by a negative quantity $K_{N}leq b<0$
In $N(k)$-contact metric manifolds and/or $(k,mu)$-manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with $V$ pointwise collinear with the structure vector field $xi $ are studied.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
