Curvature estimates of a spacelike graph in a Lorentzian product space (Q6619481)

Let \(f:\overline{B}_{x_0}(R)\subseteq M\to\mathbb{R}\) be a \(C^2\)-function defined on a closed geodesic ball \(\overline{B}_{x_0}(R)\) centered at \(x_0\in M\) with radius \(R\), where \((M,\left\langle\, ,\,\right\rangle_M)\) is a Riemannian manifold of dimension \(n\). In the spirit of Heinz curvature estimates, this paper is concerned with upper estimates of the minimum of \(|K|\), \(|H|\) and \(|B|\), where \(K\), \(H\), and \(|B|\) are, respectively, the scalar curvature, mean curvature and the norm of the second fundamental form of the spacelike graph \(\Gamma_f:=\{(x,f(x))\in \overline{B}_{x_0}(R)\times\mathbb{R};~||\nabla f||<1\}\) inside the Lorentzian product space \((M\times\mathbb{R},\left\langle\, ,\,\right\rangle_M-dt^2)\).\N\NLet \(c:=\min_{\overline{B}_{x_0}(R)}\, K_M\), \(C:=\max_{\overline{B}_{x_0}(R)}\, K_M\) and \(\xi:=||\nabla f(x_1)||\), where \(K_M\) is the scalar curvature of \(M\) and \(x_1\in \partial \overline{B}_{x_0}(R)\) such that \(f(x_1)=\max_{\partial \overline{B}_{x_0}(R)}f\). Furthermore, let \(\mu_{c,R}\) be the function\N\[\N\mu_{c,R}(s):=\begin{cases}~\frac{s}{\sqrt{1-s^2}}\cdot\sqrt{-c}\coth(\sqrt{-c} R)~& c<0 \\\N\frac{s}{R\sqrt{1-s^2}}~& c\geq 0, \end{cases}\N\]\Nfor \(0\leq s <1\).\N\NIn this setting, the author proves the following estimates:\N\begin{align*}\N\min_{\Gamma_f} |H| & \leq n\, \mu_{c,R}(\xi), \\\N\min_{\Gamma_f} |K| &\leq 2(2n-1)(n-1)\, \mu_{c,R}(\xi)^2+n(n-1)|C|, \text{ and} \\\N\min_{\Gamma_f}|B|&\leq 3(n-2)\, \mu_{c,R}(\xi),\N\end{align*}\Nwhere for the last estimate it is additionally assumed that the Ricci-curvature of \(\Gamma_f\) satisfies \(\operatorname{Ric}<(n-1)c\).\N\NMoreover, the paper provides Poincaré-Sobolev inequalities for the case of \(M\) being compact, relating the Riemannian Poincaré inequality with the mean curvature of \(\Gamma_f\) using the divergence theorem.