Scalar curvature estimates for compact symmetric spaces

Abstract

We establish extremality of Riemannian metrics g with non-negative curvature operator on symmetric spaces M=G/K of compact type with rk(G)-rk(K)\le 1. Let g' be another metric with scalar curvature k', such that g'\ge g on 2-vectors. We show that k'\ge k everywhere on M implies k'=k. Under an additional condition on the Ricci curvature of g, k'\ge k even implies g'=g. We also study area-non-increasing spin maps onto such Riemannian manifolds.Comment: 13 pages, LaTeX, uses amsar

Dirac and Plateau billiards in domains with corners

“…of depth 0) manifolds with positive scalar curvature, in particular for most (all?) compact Riemannian symmetric spaces X, see [41], [32], [12], [13], [34], [19] which is proved with Dirac operators.…”

“…Remarks on Pure Edge Singularities. The K-area inequalities and the related extremality/rigidity results for closed for Riemannian manifolds (X, g) can be expressed in terms of the size/shape of this X with the metric scal(g) ⋅ g, where, observe this metric is invariant under scaling of g. (Compare [41], [32], [33], [12].) Namely, the suitable for the present purpose area extremality of an X says that if another manifold, say (X ′ , g ′ ) admits a map f ∶ X → X ′ of positive degree than this f can not be strictly area decreasing with respect to the metrics scal(g) ⋅ g and scal(g ′ ) ⋅ g ′ .…”

Vafa-Witten Estimates for Compact Symmetric Spaces

“…Let g be a symmetric metric, and let D denote the corresponding Dirac operator on M . Ifḡ is another metric withḡ ≥ g on T M andD is the corresponding Dirac operator, thenIn the case of equality, we haveḡ = g.For an arbitrary Riemannian metricḡ such that c 2ḡ ≥ g for some suitable positive constant c 2 , the theorem impliesWe combine the methods of [9] and [4] with related estimates in [7]. In particular, we compareD to an operatorD 1 with nonvanishing kernel acting on the same sections asD 0 = D ⊗ id R k .…”

“…We use Parthasarathy's formula to compute λ 1 (D 2 ), and we exhibit a similar formula to estimate D 1 −D . Both formulas give the same value for g =ḡ.To prove thatD 1 has a kernel, we use the invariance of the Fredholm index if rk H = rk G. If rk H = rk G − 1 we use the invariance of the mod-2-index as in [7]. Unfortunately, both approaches fail if rk G− rk H ≥ 2.…”

“…To prove thatD 1 has a kernel, we use the invariance of the Fredholm index if rk H = rk G. If rk H = rk G − 1 we use the invariance of the mod-2-index as in [7]. Unfortunately, both approaches fail if rk G− rk H ≥ 2.…”