A splitting theorem for capillary graphs under Ricci lower bounds

Recent rigidity results for graphs with prescribed mean curvature

“…However, as far as we know there is still no attempt to study the equivalent of the Berestycki-Caffarelli-Nirenberg conjecture for (101). We here comment on the very recent [36], where we address this conjecture for constant , proving the following splitting theorem for capillary graphs on manifolds with non-negative Ricci curvature. The result also relates to the work in progress [57], where the authors study the problem for general , with emphasis on bistable nonlinearities.…”

“…In case ( ), then * < ∞ and hence is bounded from above. Moreover, > 0 on ( * , * ), thus (36) admits no constant solutions. The absence of non-constant solutions follows from > 0 by applying again Theorem 2.5.…”

“…The above method allows for generalizations to the CMC case, obtained in the very recent [36], that apply to the rigidity of capillary graphs over unbounded regions. We detail the application in the next section.…”

“…[36]). Let be a complete manifold of dimension ≥ 2 satisfying Ric ≥ −( − 1)2 on for some constant ≥ 0.Let Ω ⊆ be an open subset and let ∈ ∞ (Ω) be a positive solution of div √ on Ω, for some ∈ ℝ.…”