A generic result on Weyl tensor

Anna Maria Micheletti, Angela Pistoia

DOI: http://dx.doi.org/10.12775/TMNA.2018.046

Abstract


Let $M$ be a connected compact $C^\infty$ manifold of dimension $n\ge4$ without boundary. Let $ \mathcal{M}^k$ be the set of all $C^k$ Riemannian metrics on $M$. Any $g\in\mathcal{M}^k$ determines the Weyl tensor $$ \mathcal W^g\colon M\to \mathbb R^{4n},\qquad \mathcal W^g(\xi):=(W^g_{ijkl}(\xi))_{i,j,k,l=1,\dots,n}.$$ We prove that the set $$\mathcal{A}:=\big\{g\in \mathcal{M}^k : |\mathcal W^g(\xi)|+|D \mathcal W^g(\xi)|+|D^2 \mathcal W^g(\xi)|> 0\ \hbox{for any}\ \xi\in M\big\}$$ is an open dense subset of $\mathcal{M}^k$.

Keywords


Weyl tensor; Yamabe problem; generic result

Full Text:

PREVIEW FULL TEXT

References


T. Aubin, Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), 269–296.

S. Brendle, Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 21 (2008), no. 4, 951–979.

S. Brendle and F.C. Marques, Blow-up phenomena for the Yamabe equation. II, J. Differential Geom. 81 (2009), no. 2, 225–250.

M.A. Khuri, F.C. Marques and R.M. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009), no. 1, 143–196.

J. Lohkamp The Higher Dimensional Positive Mass Theorem II, arXiv:1612.07505.

F.C. Marques, Compactness and non-compactness for Yamabe-type problems, Contributions to Nonlinear Elliptic Equations and Systems, Progr. Nonlinear Differential Equations Appl. vol. 86, Birkhäuser/Springer, 2015, pp. 121–131.

A.M. Micheletti and A. Pistoia, Generic properties of critical points of Weyl tensor, Adv. Nonlinear Stud. 17 (2017), no. 1, 99–109.

M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1972), 247–258.

D. Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar curvature equation, Comm. Anal. and Geom. 1 (1993), 347–414.

F. Quinn, Transversal approximation on Banach manifolds, Global Analysis, 1970 (Proc. Sympos. Pure Math., Vol. XV, Berkeley, California, 1968), Amer. Math. Soc., Providence, R.I., pp. 213–222.

J.C. Saut and R. Temam, Generic properties of nonlinear boundary value problems, Comm. Partial Differential Equations 4 (1979), no. 3, 293–319.

R.M. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geometry 20 (1984), 479–495.

R.M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations, Lecture Notes in Mathematics, Springer–Verlag, New York, vol. 1365, 1989.

R.M. Schoen, Notes from graduates lecture in Stanford University, 1988. http://www.math.washington.edu/pollack/research/Schoen-1988-notes.html.

R.M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311–320.

R.M. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 76 (1979), 65–45.

N.S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274.

K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059–1078.

H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 21–37.

E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 402 (1981), 80–381.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism