Remarks on the Rigidity and Stability of Minimal Submanifolds

Sakaki, Makoto

doi:10.2307/2047437

Cited by 3 publications

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“…As M r has nonnegative Ricci curvature, it follows from the comparison theorem that R-m V\2R) is bounded from above. Thus we find that the volume of M is finite by choosing R sufficiently large in (17). Then ^1(M)=0, which implies the instability of M by Corollary 1.2.…”

Section: Theorem 23 Let M Be An M-dimensional Complete Riemannian Manifold Conformally Equivalent To a Bounded Open Domain In R Mmentioning

confidence: 66%

“…In our previous paper [17] we have the following inequality: (5) and the Gauss equation \A\ 2 =m(m-1)-S, we have (6) 0 for some Ci>0. We choose the function ψ as follows:…”

Section: Preliminariesmentioning

confidence: 98%

“…The first integral of the right hand side of ( 16) is an conformal invariance, so we may compute it on M f . Therefore, we have for some (17) Here V\r) is the volume of B(r) measured on M' and V(r) is the volume of B(r) measured on M, where B{r) is defined as (3) except ρ(p). As M r has nonnegative Ricci curvature, it follows from the comparison theorem that R-m V\2R) is bounded from above.…”

Section: Theorem 23 Let M Be An M-dimensional Complete Riemannian Manifold Conformally Equivalent To a Bounded Open Domain In R Mmentioning

confidence: 99%

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Nonexistence of some complete stable minimal submanifolds

Sakaki¹

1988

Kodai Math. J.

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Simons [20] proved that there is no closed stable minimal submanifold in the unit sphere, and that there is no closed stable minimal hypersurface in any Riemannian manifold with positive Ricci curvature. Complete stable minimal surfaces in 3-dimensional Riemannian manifolds with nonnegative scalar curvature were discussed by Fischer-Colbrie and Schoen [5], and complete stable minimal hypersurfaces in the Euclidean space were discussed by do Carmo and Peng [3]. In Section 2 of this paper we deal with the nonexistence problem for some complete stable minimal submanifolds in the unit sphere and some complete stable minimal hypersurfaces in a Riemannian manifold with nonnegative Ricci curvature. In Section 3 we consider an application of the nonexistence argument to the free boundary problem of minimal surfaces.The examples of complete non-compact minimal surfaces in the 3 and 5-dimensional unit sphere are obtained by use of the method in [9]. In [6] and [14] uncountably many examples of complete non-compact minimal hypersurfaces in the unit sphere are constructed.In this paper all manifolds are smooth, connected, and have dimensions not less than 2. We shall use the same < , > to denote the inner products on fibers of vector bundles of Riemannian manifolds. We denote by λ λ (M) the greatest lower bound of the spectrum of the Laplacian of a Riemannian manifold M.The author would like to express his hearty thanks to Prof. S. Tanno for his constant encouragement and advice, and to the referee for his useful comments.

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Section: Theorem 23 Let M Be An M-dimensional Complete Riemannian Manifold Conformally Equivalent To a Bounded Open Domain In R Mmentioning

confidence: 66%

“…In our previous paper [17] we have the following inequality: (5) and the Gauss equation \A\ 2 =m(m-1)-S, we have (6) 0 for some Ci>0. We choose the function ψ as follows:…”

Section: Preliminariesmentioning

confidence: 98%

Section: Theorem 23 Let M Be An M-dimensional Complete Riemannian Manifold Conformally Equivalent To a Bounded Open Domain In R Mmentioning

confidence: 99%

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Nonexistence of some complete stable minimal submanifolds

Sakaki¹

1988

Kodai Math. J.

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“…where we use the Gauss equation for the second equality. In [15] we showed that M by applying Theorem 2 to a e (1/2, 2/3]. Similarly, we may use Corollaries 1, 2 and 3 to estimate the stability of a domain of infinite area (cf.…”

Section: Lemma Let F: M^n N (A) Be a Minimal Immersion Of A 2-dimensi...mentioning

confidence: 95%

Upper bounds for the index of minimal surfaces

Sakaki¹

1990

Tohoku Math. J. (2)

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Introduction. A minimal submanifold in a Riemannian manifold is a critical point of the volume functional. Therefore, problems on the index arise naturally. Here the index in the sense of Morse is defined to be the number of negative eigenvalues of the Jacobi operator corresponding to the second variation.In this paper we obtain upper bounds for the index of a compact domain with boundary on a minimal surface in an Hadamard manifold or a space form, where an Hadamard manifold is a simply connected complete Riemannian manifold of nonpositive curvature. In [2] Berard and Besson obtained an upper bound for the index of a compact domain with boundary on a minimal submanifold of dimension greater than 2 in an Hadamard manifold. We note that their method does not apply in dimension

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“…The author could not show the inequality (4.5) only from that $f_{11}^{B}+f_{22}^{B}=0$ and $f_{12}^{B}=f_{21}^{B}$ (cf. [9]).…”

Section: Stability Of Harmonic Mapsmentioning

confidence: 99%