A spinorial analogue of Aubin's inequality

Details A spinorial analogue of Aubin's inequality A spinorial analogue of Aubin's inequality

2003-08-12

arxiv.org/abs/math/0308107v6

Mathematics

Differential Geometry

Title changed, introduction modified, main result has changed, applications added

Scientific paper

Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$. For any metric $\tilde g$ conformal to $g$, we denote by $\tilde\lambda$ the first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show that $$\inf_{\tilde{g} \in [g]} \tilde\lambda \Vol(M,\tilde g)^{1/n} \leq (n/2) \Vol(S^n)^{1/n}.$$ This inequality is a spinorial analogue of Aubin's inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case $n \geq 3$ and in the case $n = 2$, $\ker D=\{0\}$. Our proof also works in the remaining case $n=2$, $\ker D\neq \{0\}$. With the same method we also prove that any conformal class on a Riemann surface contains a metric with $2\tilde\lambda^2\leq \tilde\mu$, where $\tilde\mu$ denotes the first positive eigenvalue of the Laplace operator.

Affiliated with

Also associated with

No associations

Rating

A spinorial analogue of Aubin's inequality does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A spinorial analogue of Aubin's inequality, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A spinorial analogue of Aubin's inequality will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-210717