A Characterization of the Unit Ball by a Kähler–Einstein Potential

harmonic 1-forms on hypersurfaces with finite index

“…For a sufficiently small such that , Together with (3.17), we have for some positive constant . In what follows, as in [7], we consider any finite dimensional subspace . According to Lemma 2.9, we see that there exists an harmonic 1-form such that From (3.18), we have which implies that is bounded by a fixed constant.…”

“… Combining (3.11) and (3.13) and the assumption , we get for some positive constant . Now we choose our test function as in [7]: given , on and on . Applying this test function to (3.14), we get Letting and using the assumption that , we obtain Then, using Hölder inequality and (3.15), we conclude that Adding to both sides of (3.16), we get Again adding to both sides infers On the other hand, since satisfies the differential inequality (3.3), Lemma 2.8 implies that for some positive constant .…”

“…The first author and Lv [6] also investigated the nonexistence of nontrivial harmonic 1-form of a complete -stable hypersurface with weighted Poincaré inequality in a Riemannian manifold with sectional curvature bounded below by a nonpositive function. Most recently, without the stability assumption, Choi and Seo [7] proved the following finiteness theorem.…”

“…([7]). Let N be an -dimensional complete simply connected Riemannian manifold with sectional curvature satisfying for a nonzero constant k. Let M be an -dimensional complete noncompact minimal hypersurface with finite index in N. For , assume that Then, .…”