2018 # 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.

Help me understand this report

Search citation statements

Paper Sections

Select...

2

1

1

1

Citation Types

1

14

0

Year Published

2019

2022

Publication Types

Select...

5

2

Relationship

1

6

Authors

Journals

(15 citation statements)

(91 reference statements)

1

14

0

“…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.…”

confidence: 73%

“…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.…”

confidence: 73%

“…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.…”

confidence: 91%

“…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].…”

confidence: 64%

“…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].…”

confidence: 81%