Biharmonic hypersurfaces with constant scalar curvature in space forms

Abstract

Let M n be a biharmonic hypersurface with constant scalar curvature in a space form M n+1 (c). We show that M n has constant mean curvature if c > 0 and M n is minimal if c ≤ 0, provided that the number of distinct principal curvatures is no more than 6. This partially confirms Chen's conjecture and Generalized Chen's conjecture. As a consequence, we prove that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space E n+1 or hyperbolic space H n+1 for n < 7.

Closed Biconservative Hypersurfaces in Spheres

“…The case n = 3 corresponds with p = 1/4 and the above result was obtained in [22]. When n = 5, p = 1/2 and the associated Euler-Lagrange equation coincides with the ODE obtained in [14] when studying biconservative hypersurfaces with constant scalar curvature. We point out here that among non-CMC rotational biconservative hypersurfaces, the only ones with constant scalar curvature appear when the dimension is n = 5.…”

“…In other words, the curve γ d never passes through the pole (1/ √ ρ, 0, 0). Finally, differentiating (14) with respect to the arc length parameter, we observe that the function ψ(s) is monotonic for each half period of the curvature. Thus, the p-elastic curve γ d goes always forward and it does not cut itself in one period of its curvature, unless it gives more than one round in that period.…”

“…Then, from Proposition 4.1, the curvature of γ d is a periodic function, and we denote by ≡ (d) its period. It then follows from the parametrization (13), that γ d is closed if and only if the integral ψ(s), (14), along a natural multiple of is a natural multiple of 2π, i.e., assuming without loss of generality that the origin of the arc length parameter is s = 0, if and only if…”

Biharmonic and biconservative hypersurfaces in space forms

“…Remark 4.11. The above result holds in any space form N m+1 (c), as it is actually given in its original form in [35].…”

Linear Weingarten $$\lambda $$-biharmonic hypersurfaces in Euclidean space

“…Note that the assumption linear Weingarten is much weaker than the general geometric assumptions constant scalar curvature or constant mean curvature. Hence, Theorem 1.1 is a generalization of Corollary 1.4 in [20].…”