{"paper":{"title":"The maximal principle for properly immersed submanifolds and its applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Yong Luo","submitted_at":"2015-05-25T07:41:48Z","abstract_excerpt":"In this note we consider the Liouville type theorem for a properly immersed submanifold $M$ in a complete Riemmanian manifold $N$. Assume that the sectional curvature $K^N$ of $N$ satisfies $K^N\\geq-L(1+dist_N(\\cdot,q_0)^2)^\\frac{\\alpha}{2}$ for some $L>0, 2>\\alpha\\geq 0$ and $q_0\\in N$.\n  (i) If $\\Delta|\\vec{H}|^{2p-2}\\geq k|\\vec{H}|^{2p}$($p>1$) for some constant $k>0$, then we prove that $M$ is minimal.\n  (ii) Let $u$ be a smooth nonnegative function on $M$ satisfying $\\Delta u\\geq ku^a$ for some constant $k>0$ and $a>1$. If $|\\vec{H}|\\leq C(1+dist_N(\\cdot,q_0)^2)^\\frac{\\beta}{2}$ for some "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06555","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}