Stability of capillary hypersurfaces in a manifold with density (Q2816972)

An \((n+1)\)-dimensional Riemannian manifold \((\tilde{M}, g,fdv)\) is considered with density \(f=e^{\psi}\), with boundary \(\partial \tilde{M}\) of unit outward normal \(\bar{N}\), and an orientable immersed hypersurface \(x:M^n\to \tilde{M}\) with boundary satisfying \(x(\partial M)\subset \partial\tilde{M}\). The unit normal \(N\) of \(M\) in \(\tilde{M}\) is chosen to point inwards to \(T\), where \(T\) is the part of \(\tilde{M}\) such that its boundary is \(x(M) \cup \Omega\), with \(\Omega=x(\partial {M})\subset \partial \tilde{M}\). Additionally, two unit vectors are defined, the conormal \(\nu\) of \(\partial M\) in \(M\) and the conormal of \(\partial M\) \(\bar{\nu}\) in \(\Omega\), and the contact angle \(\theta\) between them. The hypersurface \(x:M\to \tilde{M}\) is a capillary if for some \(\theta_0\in (0,\pi)\) it is a critical point of the energy functional on de \(f\)-area \(A_f\) NEWLINENEWLINE\[NEWLINE E_f(t)=A_f(M(t))-\cos\theta_0 A_f(\Omega(t)),NEWLINE\]NEWLINE NEWLINEwhere \(X(t,p)\) is a \(f\)-volume one-parameter variation of \(x\) with \(X_t(\partial M)\subset \partial\tilde{M}\), that is, the \(f\)-volume \(V_f(t)=\int_{[0,t]\times M}X^*dv_f\) is constant on \(t\). By the first variation formula for \({E'}_f(0)\) and \({V'}_f(0)\), this means that \(M\) has constant \(f\)-mean curvature \(H_f\) and makes constant contact angle \(\theta=\theta_0\) with \(\partial\tilde{M}\). Capillarity has the physical meaning of the contact of two materials in absence of gravity, namely \(M\) describes the interface of the air and some incompressible liquid drop \(T\) inside a container \(B\) represented by \(\tilde{M}\). If \(\theta_0=\pi/2\) the capillary hypersurface is said to have free boundary. A capillary hypersurface is stable (resp. strongly stable) if \({E''}_f(0)\geq 0\) for all \(\phi\in \mathcal{F}\) (resp. for all \(\phi\in C^{\infty}(M)\)), where the second variation is given by NEWLINENEWLINE\[NEWLINE \partial^2E_f(\phi):={E''}_f(0)=-\int_M (\Delta_{M,f}\phi +(|\sigma|^2+ \mathrm{Ric}_f(N,N))\phi)\phi da_f + \int_{\partial M}(\frac{\partial \phi}{\partial \nu} -q\phi)\phi ds_fNEWLINE\]NEWLINE NEWLINEwhere \(\phi=\langle Y,N\rangle\) and \(Y\) is the vector variation of \(X\), an element of \(\mathcal{F}=\{\phi \in H^1(M): \int_M\phi da_f=0\}\), \(\mathrm{Ric}_f= \mathrm{Ric} -D^2\psi\) the \(f\)-Ricci tensor of \(\tilde{M}\), \(\sigma\) is the second fundamental form of \(M\) in \(\tilde{M}\), \(q=\frac{1}{\sin \theta}\Pi(\bar{\nu}, \bar{\nu})+\cot\theta\, \sigma(\nu,\nu)\), where \(\Pi\) is the second fundamental form of \(\partial \tilde{M}\) in \(\tilde{M}\), and \(\Delta_{M,f}\phi= \Delta_M\phi + \langle \nabla \psi, \nabla \phi\rangle\). This second variation formula is specialized for the case where \(\tilde{M}\) is a geodesic ball \(B^{n+1}(d)\) of radius \(d\) of an \((n+1)\)-dimensional space form with a radial density \(f=e^{\psi}\), i.e., \(\psi=\psi(r)\), and \(x:M\to B^{n+1}(d)\) is a stable free boundary \(f\)-minimal hypersurface. Using suitable test functions \(\phi\) the authors conclude in Theorems 1, 2 and 3 that \(M\) must be totally geodesic in the following cases: in the Euclidean case for \(d=1\) and \(\psi' -\psi'' r\geq 0\), in the hyperbolic space with any \(d\) and \(\psi' \cosh r -\psi''\sinh r\geq 0\), and in the standard \((n+1)\)-sphere with \(d<\pi\) with any \(d\) and \(\psi' \cos r -\psi''\sin r\geq 0\).NEWLINENEWLINEThey also prove in Theorem 4 for the Euclidean unit ball with \(\psi''\leq 0\) that any capillary hypersurface in \(B^{n+1}\) defining \(T\) symmetric with respect to the origin is unstable. The same instability holds for \(x\) with free boundary and replacing the symmetric assumption on \(T\) by the condition that the Gauss map satisfies \(\int_MNda_f=0\). These results generalize the known case \(f=1\).NEWLINENEWLINEIn the last Theorem 6, a manifold \(\tilde{M}\) of dimension 3 with \(f\)-mean convex boundary is considered. Assuming \(M\) is a compact oriented capillary surface and strongly stable, by choosing \(\phi=1/\sqrt{f}\) in the expression for the second variation described in terms of the Perelman scalar curvature \(S_f\), the \(f\)-mean curvature \(H_f\) and the Gauss curvature \(K\) of \(M\), and by applying the Gauss-Bonnet theorem, the authors arrive at an inequality on the Euler characteristic of \(M\), \(\chi(M)\geq \cot \theta H_f |\partial M|\), generalizing several known results for \(M\) with free boundary, or when \(M\) without boundary, and also the case of constant density.