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Register a new userAsymptotic Dirichlet problem for A-harmonic and minimal graph equations in Cartan-Hadamard manifolds
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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.
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We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.
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.
We show, by modifying Borbelys example, that there are $3$-dimen-sional Cartan-Hadamard manifolds $M$, with sectional curvatures $le -1$, such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.
We state and prove a Chern-Osserman Inequality in terms of the volume growth for minimal surfaces properly immersed in a Cartan-Hadamard manifold N with sectional curvatures bounded from above by a negative quantity.
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.
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