Variants on Alexandrov reflection principle and other applications of maximum principle

$f$-extremal domains in hyperbolic space

“…This result can be compare to Theorem C in [21] by Sa Earp and Toubiana. We can also think to Theorem A in [6] by do Carmo and Lawson about the geometry of constant mean curvature hypersurfaces in H n .…”

“…The study of f -extremal domains in H n already appears in the work of Espinar and Mao [8]. They use the fact that the hyperbolic space can be compactified by its ideal boundary ∂ ∞ H n ; so f -extremal domains can be studied in terms of their trace on ∂ ∞ H n , one of their results states that a f -extremal domain whose trace on ∂ ∞ H n is at most one point is either a geodesic ball or a horoball (hypothesis (H2) is not need in this result), this generalizes, to any f , results by Molzon [16] and Sa Earp and Toubiana [21].…”

A Survey on Alexandrov-Bernstein-Hopf Theorems

“…Furthermore, one can extend the Alexandrov Theorem to embedded hypersurfaces of R n+1 having positive constant scalar curvature, or such that any other symmetric function of the principal curvatures is a positive constant [52]. Alexandrov type results where obtained for Weingarten surfaces [11], [58] and for constant mean curvature surfaces bounded by convex curves in space forms [12], [59], [60].…”