Showing papers by "Stefano Montaldo published in 2011"
TL;DR: Chen et al. as mentioned in this paper showed that a proper biharmonic submanifold is of 1-type or 2-type if and only if it has constant mean curvature.
Abstract: We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we determine all biharmonic surfaces with parallel normalized mean curvature vector field in $\mathbb{S}^n$.
Then we investigate, for (not necessarily compact) proper biharmonic submanifolds in $\mathbb{S}^n$, their type in the sense of B-Y. Chen. We prove: (i) a proper biharmonic submanifold in $\mathbb{S}^n$ is of 1-type or 2-type if and only if it has constant mean curvature ${\mcf}=1$ or ${\mcf}\in(0,1)$, respectively; (ii) there are no proper biharmonic 3-type submanifolds with parallel normalized mean curvature vector field in $\mathbb{S}^n$.
54 citations
TL;DR: In this paper, the geodesics on an invariant surface of a 3D Riemannian manifold were studied and the main results were the characterization of geodesic orbits, Clairaut's relation and its geometric interpretation in some remarkable three dimensional spaces, the local description of the geodeics, and the explicit description of geodeic curves on a surface with constant Gauss curvature.
7 citations
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TL;DR: In this article, a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps is described, which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries.
Abstract: In this paper we describe a 1-dimensional variational approach to the analytical construction of equivariant biharmonic maps. Our goal is to provide a direct method which enables analysts to compute directly the analytical conditions which guarantee biharmonicity in the presence of suitable symmetries. In the second part of our work, we illustrate and discuss some examples. In particular, we obtain a 1-dimensional stability result, and also show that biharmonic maps do not satisfy the classical maximum principle proved by Sampson for harmonic maps.
3 citations
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TL;DR: In this paper, it was shown that a map is subelliptic biharmonic if and only if its vertical lift to the (total space of the) canonical circle bundle is a bi-harmonic map with respect to the Fefferman metric.
Abstract: We study subelliptic biharmonic maps, i.e. smooth maps from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of a certain bienergy functional. We show that a map is subelliptic biharmonic if and only if its vertical lift to the (total space of the) canonical circle bundle is a biharmonic map with respect to the Fefferman metric.