Nonnegative Scalar Curvature and Area Decreasing Maps

Let ( M, g ) be a noncompact complete spin Riemannian manifold of even dimension n, with k denote the associated scalar curvature. Let f : M → S(1) be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if k ≥ n(n − 1) on the support of df , then inf ( k ) < 0. This answers a question of Gromov. We use a simple deformation of the Dirac operator to prove the result. The odd dimensional analogue is also presented.


Introduction
It is well-known that starting with the famous Lichnerowicz vanishing theorem [5], Dirac operators have played important roles in the study of Riemannian metrics of positive scalar curvature on spin manifolds (cf. [3] and [4]). A notable example is Llarull's rigidity theorem [6] which states that for a compact spin Riemannian manifold M, g T M of dimension n such that the associated scalar curvature k T M verifies that k T M ≥ n(n − 1), any (non-strict) area decreasing smooth map f : M → S n (1) of nonzero degree is an isometry.
Recently, Gromov states in [2, p. 45] a noncompact extension of the Llarull theorem. Namely, if M, g T M is an n dimensional noncompact complete spin Riemannian manifold, f : M → S n (1) a smooth (non-strict) area decreasing map (which is locally constant near infinity) of nonzero degree such that k T M ≥ n(n − 1) on the support of df , then inf k T M ≤ 0. (1.1) The argument used by Gromov for (1.1) relies on the relative index theorem of Gromov-Lawson [3], which depends on the positivity of k T M near infinity. Gromov then raises the question that whether the inequality in (1.1) can actually be made strict.
The purpose of this short note is to provide a positive answer to this question when n is even. That is, (1.1) can indeed be improved to inf k T M < 0. When n is odd, we improve (1.1) to inf k T M < 0 under the condition that k T M > n(n − 1) on the support of df .
The main idea of the proof, similar to [8, equation (1.11)], is to deform the involved twisted Dirac operator (constructed as in [6]) on M by a suitable endomorphism of the twisted vector bundle (cf. (2.8)). The deformed Dirac operator turns out to be invertible near infinity, and one can then apply the relative index theorem to complete the proof. This paper is a contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Gromov.html

The main results and their proofs
This section is organized as follows. In Section 2.1, we restate the main results of this paper. In Sections 2.2 and 2.3, we prove the main results stated in Section 2.1.

The main results
Let M, g T M be an n-dimensional noncompact spin complete Riemannian manifold. Let k T M be the associated scalar curvature. Let S n (1) be the standard n-dimensional unit sphere carrying its canonical metric. Following [3], a smooth map f : M → S n (1) is called area decreasing if for any two form α ∈ Ω 2 S n (1) , f * α ∈ Ω 2 (M ) verifies that |f * α| ≤ |α|. (2.1) We now assume that f : M → S n (1) is a smooth area decreasing map such that it is locally constant near infinity. That is, it is locally constant outside a compact subset K ⊂ M . We also assume that Let df : T M → T S n (1) be the differential of f . The support of df is defined to be Supp(df ) = {x ∈ M : df (x) = 0}.
The main result of this short note can be stated as follows.
Theorem 2.1. Under the above assumptions, if n is even and   [4]). And we use similar notation for S n (1). Following [6], let E = E + ⊕ E − be the Z 2 -graded Hermitian vector bundle over M carrying the pull-back Hermitian connection Let D E : Γ S(T M ) ⊗E → Γ S(T M ) ⊗E be the canonically defined (twisted by E) Dirac operator (cf. [4]). 1 Then one has the canonical splitting Let v * : S − T S n (1) → S + T S n (1) be the adjoint of v with respect to the Hermitian metrics on S ± (T S n (1)). Let V : S T S n (1) → S T S n (1) be the self-adjoint odd endomorphism defined by Then one has The existence of ϕ is clear.
where c(·) is the notation for the Clifford action, [·, ·] is the notation for the supercommutator in the sense of [7], and we identify dϕ with the gradient of ϕ. Let e 1 , . . . , e n be an orthonormal basis of T M, g T M . By the definition of the Dirac operator, c(e i )∇ e i , one has (cf. [8]) Since by definition ϕ = 1 on M \K, while f is locally constant on M \K, from (2.9) and (2.10) we see that the following identity holds on M \ K, (2.11) 1 Here " ⊗" is the notation for the Z2-graded tensor product (cf. [7]). 2 In view of [7], one may regard D E ε as a "super" Dirac operator.
As V | f (M \K) is invertible, from (2.11), one sees that there is a constant a > 0 such that for any s ∈ Γ S(T M ) ⊗E supported in a compact subset of M \ K, one has D E ε s ≥ εa s .

Remark 2.3.
In view of (2.12) and [1, Section 6 4 5 ], one sees that the above proof fits with Gromov's suggestion in [2, p. 45] that one may use the Callias type index arguments to deal with (2.4). Also, from the above proof one sees that (2.3) can be weakened to

Proof of Theorem 2.2
In view of Remark 2.3, we will state and prove the following refined version of Theorem 2.2.
Theorem 2.4. Let M, g T M be a noncompact spin complete Riemannian manifold of odd dimension n, and f : M → S n (1) be a smooth map which is locally constant near infinity and of nonzero degree. If we assume that the scalar curvature k T M of g T M verifies that Proof . For any R > 0, let S 1 (R) be the round circle of radius R, with the canonical metric dt 2 . Let M × S 1 (R) be the complete Riemannian manifold of the product metric g T M ⊕ dt 2 . Following [6], we consider the chain of maps (1) is the standard shrinking map, and h is a suspension map of degree one such that |dh| ≤ 1. (1) be the Clifford action of X. Let V : S T S n+1 (1) → S T S n+1 (1) be defined as in (2.6). Then (2.7) holds and V is invertible on S n+1 (1) \ {p}. In particular, there is δ > 0 such that The analogue of (2.9) now takes the form (2.28) From (2.26), (2.27) and (2.28), one sees that for any ε > 0, there exist R 0 > 0 and b > 0 such that when R ≥ R 0 , for any s ∈ Γ S(T M R ) ⊗E R supported in a compact subset of (M \ K) × S 1 (R), one has One can then apply the relative index theorem [3] to D E R ε,+ and get as in (2.13) that Let e 1 , . . . , e n+1 be an orthonormal basis of T M R . The Lichnerowicz formula (2.14) now takes the form where π R : M × S 1 (R) → M denotes the natural projection. By [6, p. 68], one has at any (x, y) ∈ M × S 1 (R) that c(e i )c(e j )R E R (e i , e j ) ≥ − n(n − 1) 4 ∧ 2 (df (x)) + |df (x)|O 1 R .