The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

Ivanov, Stefan; Vassilev, Dimiter

doi:10.1016/j.na.2015.06.024

Cited by 15 publications

(19 citation statements)

References 198 publications

(369 reference statements)

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“…where R f (Z) = ∇df (ξ, JZ), see [16,Section 7.1] and references therein but note the opposite sign of the sub-Laplacian, we obtain the next identity (3.6) 1 2 (∂ t − ∆)(uw) = − |∇ 2 f | 2 − Ric(∇f, ∇f ) − 2A(∇f, ∇∇f ) − 4R f (∇f ) u.…”

Section: The Cr Casementioning

confidence: 78%

“…Since (2.10) still holds, working as in the qc case we compute M R F (∇f )u V ol η in two ways [7,Lemma 4] and [14,Lemma 8.7] following the exposition [16]. On the other hand, using again (3.7) but now we integrate and then use integration by parts we have At this point, exactly as in the qc case, we subtract (3.9) and three times formula (3.8) from (3.6), which gives…”

Section: The Cr Casementioning

confidence: 99%

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Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds

Vassilev

2017

Pacific J. Math.

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Section: The Cr Casementioning

confidence: 78%

Section: The Cr Casementioning

confidence: 99%

Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds

Vassilev

2017

Pacific J. Math.

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“…The proof of the local equivalence statement in Theorem 1.1 follows, for more details see [7,Theorem 1.2] in the case of positive qc-scalar curvature, the paragraph after [14,Lemma 8.6] in the zero qc-scalar curvature case, while the negative qc-scalar curvature case follows analogously. The global result in the case of a compact manifold is achieved by a monodromy argument and Liouville's theorem [14,Theorem 8.5], [3]. Below is an argument using that in our case Biquard's connection is an affine connection with parallel torsion and parallel curvature, hence we can invoke the results in [15,Chapter VI].…”

Section: Appendixmentioning

confidence: 92%

Non-Umbilical Quaternionic Contact Hypersurfaces in Hyper-Kähler Manifolds

Ivanov

Minchev

Vassilev

2017

International Mathematics Research Notices

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We show that any compact quaternionic contact (abbr. qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-Kähler manifold is endowed with an involutive 7-dimensional distribution whose integral leafs are locally qc-conformal to the standard 3-Sasakian sphere. in [2,4], every real analytic qc structure is the conformal infinity of a unique (asymptotically hyperbolic) quaternionic-Kähler metric defined in a neighborhood of the qc structure. Similar to the CR case the question of embedded quaternionic contact hypersurfaces is a natural one, but in contrast to the CR case it imposes a rather strong conditions on the hypersurface. The situation has the flavor of the Kähler versus the hyper-Kähler case. As well known any complex submanifold of a Kähler manifold is a Kähler manifold and a Kähler metric is locally given by a Kähler potential. In contrast, a hyper-complex manifold of a hyper-Kähler manifold must be totally geodesic and (in general) there is no hyper-Kähler potential (the structure is rigid). This suggests that we can expect that there are few quaternionic contact hypersurfaces in a hyper-Kähler manifold. Indeed, we showed in [9] that given a connected qc-hypersurface M in the flat quaternion space H n+1 , then, up to a quternionic affine transformation of H n+1 , M is contained in one of the following three hyperquadrics (the 3-Sasakain sphere, the hyperboloid and the quaternionic Heisenberg group):(1.1) (i) |q 1 | 2 +· · ·+|q n | 2 +|p| 2 = 1, (ii) |q 1 | 2 +· · ·+|q n | 2 −|p| 2 = −1, (iii) |q 1 | 2 +· · ·+|q n | 2 +Re(p) = 0, Date: September 24, 2018. 1991 Mathematics Subject Classification. 58G30, 53C17.

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“…Once we know this new notion of curvature, we can define the qc-Yamabe problem: determinate if there exists a metric e 2f g in the conformal class [g] with constant qc-scalar curvature. This problem has been solved in great generality, see [IV2] for a survey. In our case M = S 4n−1 , g = σ sR and for every i η i (·) = σ(ξ i , ·).…”

Section: The Biquard Connection and The Qc-scalar Curvaturementioning

confidence: 99%

Inverse mean curvature flow in quaternionic hyperbolic space

Pipoli

2018

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub-Riemannian metric on the sphere defined on a codimension 3 distribution. Finally we show that there exists a family of examples such that the qc-scalar curvature of this sub-Riemannian limit is not constant.MSC 2010 subject classification 53C17, 53C40, 53C44.

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The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

Cited by 15 publications

References 198 publications

Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds

Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds

Non-Umbilical Quaternionic Contact Hypersurfaces in Hyper-Kähler Manifolds

Inverse mean curvature flow in quaternionic hyperbolic space

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