The maximum principle for hypersurfaces with vanishing curvature functions

Hounie, Jorge; Leite, Maria Luiza

doi:10.4310/jdg/1214456216

Cited by 42 publications

(34 citation statements)

References 4 publications

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“…For a proof of (i), see [3], while for (ii), see [10]. From the above proposition, the requirements on p and rank(A) in the main theorems ensure ellipticity.…”

Section: Lemma 21 Let {E I } Be the Principal Directions Associatedmentioning

confidence: 93%

Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space

Impera

Mari

Rigoli

2010

Proc. Amer. Math. Soc.

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Abstract. Let f : M → R m+1 be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors to analyze the stability of the differential operator L r associated with the rth Newton tensor of f . This appears in the Jacobi operator for the variational problem of minimizing the r-mean curvature H r . Two natural applications are found. The first one ensures that under a mild condition on the integral of H r over geodesic spheres, the Gauss map meets each equator of S m infinitely many times. The second one deals with hypersurfaces with zero (r + 1)-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces f * T p M , p ∈ M , fill the whole R m+1 .

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“…For a proof of (i), see [3], while for (ii), see [10]. From the above proposition, the requirements on p and rank(A) in the main theorems ensure ellipticity.…”

Section: Lemma 21 Let {E I } Be the Principal Directions Associatedmentioning

confidence: 93%

Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space

Impera

Mari

Rigoli

2010

Proc. Amer. Math. Soc.

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“…Since we are assuming that the profile curve is parametrized by the arc length, the rank of the second fundamental form of the immersion ξ : M p+q+1 → R p+q+2 is greater than min{p, q} ≥ r > r − 1. Therefore, the associated Jacobi operator J r−1 is an elliptic operator (see [9]). …”

Section: Classification Of the Invariant Hypersurfacesmentioning

confidence: 99%

O(p + 1) × O(q + 1)-invariant (r – 1)-minimal hypersurfaces in Euclidean space

Sousa

2010

Advg

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“…, k i n are the eigenvalues of A = −dg, where g: M n → S n (1) is the Gauss map of the hypersurface. Reilly showed in [8] that orientable hypersurfaces with S r +1 = 0 are critical points of the functional A breakthrough in the study of these hypersurfaces occurred in 1995 when Hounie and Leite [6,7] found conditions for the linearization of the partial differential equation S r +1 = 0 to be an elliptic equation. This linearization involves a second order differential operator L r (see the definition of L r 2 in Section 2) and the Hounie-Leite conditions read as follows:…”

Section: Introductionmentioning

confidence: 98%

On Stability of Cones in Rn+1 with Zero Scalar Curvature

Barbosa

Carmo

2005

Ann Glob Anal Geom

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In this work we generalize the case of scalar curvature zero the results of Simmons (Ann. Math. 88 (1968), 62-105) for minimal cones in R n+1 . If M n−1 is a compact hypersurface of the sphere S n (1) we represent by C(M) ε the truncated cone based on M with center at the origin. It is easy to see that M has zero scalar curvature if and only if the cone base on M also has zero scalar curvature. Hounie and Leite (J. Differential Geom. 41 (1995), 247-258) recently gave the conditions for the ellipticity of the partial differential equation of the scalar curvature. To show that, we have to assume n 4 and the three-curvature of M to be different from zero. For such cones, we prove that, for n 7 there is an ε for which the truncate cone C(M) ε is not stable. We also show that for n 8 there exist compact, orientable hypersurfaces M n−1 of the sphere with zero scalar curvature and S 3 different from zero, for which all truncated cones based on M are stable. (2000): 53C42, 53C40, 49F10, 57R70. Mathematics Subject Classifications

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The maximum principle for hypersurfaces with vanishing curvature functions

Cited by 42 publications

References 4 publications

Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space

Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space

O(p + 1) × O(q + 1)-invariant (r – 1)-minimal hypersurfaces in Euclidean space

On Stability of Cones in Rn+1 with Zero Scalar Curvature

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