Local Study of Scalar Curvature of Cyclic Surfaces Obtained by Homothetic Motion of Lorentzian Circle

Abstract Classical rotating closed string are folded strings. At the folding points the scalar curvature associated with the induced metric diverges. As a consequence one cannot properly quantize the fluctuations around the classical solution since there is no complete set of normalizable eigenmodes. Furthermore in the non-critical effective string action of Polchinski and Strominger, there is a divergence associated with the folds. We overcome this obstacle by putting a massive particle at each folding point which can be used as a regulator. Using this method we compute the spectrum of quantum fluctuations around the rotating string and the intercept of the leading Regge trajectory. The results we find are that the intercepts are a = 1 and a = 2 for the open and closed string respectively, independent of the target space dimension. We argue that in generic theories with an effective string description, one can expect corrections from finite masses associated with either the endpoints of an open string or the folding points on a closed string. We compute explicitly the corrections in the presence of these masses.

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Quaternionic contact 4n + 3-manifolds and their 4n-quotients

Annals of Global Analysis and Geometry ◽

10.1007/s10455-021-09758-5 ◽

2021 ◽

Author(s):

Yoshinobu Kamishima

Keyword(s):

Scalar Curvature ◽

Lie Group ◽

Contact Structure ◽

Einstein Manifolds ◽

Hermitian Metrics ◽

Kähler Metric ◽

Formula One ◽

Quaternion Space ◽

Kahler Metric ◽

Do So

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

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Isoperimetry, Scalar Curvature, and Mass in Asymptotically Flat Riemannian 3‐Manifolds

Communications on Pure and Applied Mathematics ◽

10.1002/cpa.21981 ◽

2021 ◽

Vol 74 (4) ◽

pp. 865-905

Author(s):

Otis Chodosh ◽

Michael Eichmair ◽

Yuguang Shi ◽

Haobin Yu

Keyword(s):

Scalar Curvature ◽

Asymptotically Flat

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Almost Non-negative Scalar Curvature on Riemannian Manifolds Conformal to Tori

Journal of Geometric Analysis ◽

10.1007/s12220-021-00677-2 ◽

2021 ◽

Author(s):

Brian Allen

Keyword(s):

Scalar Curvature ◽

Riemannian Manifolds

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On far-outlying constant mean curvature spheres in asymptotically flat Riemannian 3-manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2019-0034 ◽

2020 ◽

Vol 2020 (767) ◽

pp. 161-191

Author(s):

Otis Chodosh ◽

Michael Eichmair

Keyword(s):

Scalar Curvature ◽

Mean Curvature ◽

Constant Mean Curvature ◽

Existence Results ◽

Asymptotically Flat ◽

Weingarten Surfaces ◽

Differential Geom

AbstractWe extend the Lyapunov–Schmidt analysis of outlying stable constant mean curvature spheres in the work of S. Brendle and the second-named author [S. Brendle and M. Eichmair, Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94 2013, 3, 387–407] to the “far-off-center” regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable constant mean curvature spheres that depend delicately on the behavior of scalar curvature at infinity.

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