Properly immersed submanifolds in complete Riemannian manifolds

Some open problems and conjectures on submanifolds of finite type: recent development

“…• Biharmonic submanifolds with finite total mean curvature in a Riemannian manifold of non-positive sectional curvature [148]. • Biharmonic properly immersed submanifolds in a complete Riemannian manifold with non-positive sectional curvature whose sectional curvature has polynomial growth bound of order less than 2 from below [139]. • Complete biharmonic submanifolds with finite bi-energy and energy in a non-positively curved Riemannian manifold [149].…”

Biharmonic hypersurfaces with bounded mean curvature

“…For p-biharmonic submanifolds, it is easy to see that we can get same (similar) results as in the results of biharmonic submanifolds in many cases. (For example, Corollary 3.6, 3.9 in [12], and so on.) In fact, the same argument as in Proof of Theorem 1.3 shows the following result.…”

The maximal principle for properly immersed submanifolds and its applications

“…We will prove the theorem by a contradiction argument. Here we follow Maeta's( [27]) argument by choosing new test functions. Suppose that H(x 0 ) = 0 for some…”

“…Remark 1.1. When p = 2, theorem 1.1 was proved by Maeta(see [27]). Our proof follows his argument by using the second derivatives' test to our new test functions.…”

“…In this note we will continue to consider the nonexistence of p-biharmonic submanifolds in nonpositively curved Riemannian manifold. Before mentioning our main result, we define the following notion(see [27]). Definition 1.1.…”