RIGIDITY OF NONPOSITIVELY CURVED MANIFOLDS WITH CONVEX BOUNDARY MOHAMMAD GHOMI AND JOEL SPRUCK

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RIGIDITY OF NONPOSITIVELY CURVED MANIFOLDS

WITH CONVEX BOUNDARY

MOHAMMAD GHOMI AND JOEL SPRUCK

Abstract. We show that a compact Riemannian 3-manifold Mwith strictly convex

simply connected boundary and sectional curvature K≤a≤0is isometric to a con-

vex domain in a complete simply connected space of constant curvature a, provided

that K≡aon planes tangent to the boundary of M. This yields a characteri-

zation of strictly convex surfaces with minimal total curvature in Cartan-Hadamard

3-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-

Strake. Our proof is based on a recent comparison formula for total curvature of

Riemannian hypersurfaces, which also yields some dual results for K≥a≥0.

1. Introduction

ACartan-Hadamard manifold His a complete simply connected Riemannian n-space

with sectional curvature K≤0. Greene and Wu [9,10] and Gromov [3, Sec. 5] showed

that, when n≥3, these spaces exhibit remarkable rigidity properties, analogous to

those observed earlier by Mok, Siu, and Yau [13,18] in Kähler geometry. In particular,

a fundamental result is that if Kvanishes outside a compact set C⊂ H, then His

isometric to Euclidean space Rn. More generally, if K≤a≤0on H, and K≡aon

H \ C, then K≡aon H[9, p. 734] [17]. We extend this result when n= 3:

Theorem 1.1. Let Mbe a compact Riemannian 3-manifold with nonempty C2boundary

∂M and sectional curvature K≤a≤0. Suppose that ∂M is strictly convex, each

component of ∂M is simply connected, and K≡aon planes tangent to ∂M. Then M

is isometric to a convex domain in a Cartan-Hadamard manifold of constant curvature

a. In particular, Mis diﬀeomorphic to a ball.

Strictly convex here means that the second fundamental form of ∂M is positive deﬁnite

with respect to the outward normal. For n= 3, this theorem immediately implies the

rigidity results mentioned above, by letting Mbe a geodesic ball in Hcontaining C.

Date: March 10, 2023 (Last Typeset).

2010 Mathematics Subject Classiﬁcation. Primary: 53C20, 58J05; Secondary: 53C42, 52A15.

Key words and phrases. Cartan-Hadamard manifold, Hyperbolic space, Gap theorem, Minimal total

curvature, Asymptotic rigidity, Gauss-Kronecker curvature, Bounded sectional curvature.

The research of M.G. was supported by NSF grant DMS-2202337 and a Simons Fellowship. The

research of J.S. was supported by a Simons Collaboration Grant.

arXiv:2210.05588v2 [math.DG] 8 Mar 2023

2 MOHAMMAD GHOMI AND JOEL SPRUCK

Schroeder and Strake [15] had established this result for a= 0 (and only for n= 3)

reﬁning earlier work of Schroeder and Ziller [14]. The simply connected assumption on

components of ∂M is necessary, as can be seen by considering a tubular neighborhood

of a closed geodesic in a hyperbolic manifold.

As an application of Theorem 1.1 we obtain the following characterization for strictly

convex surfaces with minimal total curvature. We say that an oriented closed (com-

pact, connected, without boundary) hypersurface Γ⊂ H is strictly convex if its second

fundamental form II is positive deﬁnite. Then Γis embedded, bounds a convex do-

main, and is simply connected [1]. The total Gauss-Kronecker curvature of Γis given

by G(Γ) := RΓdet(II), and |Γ|denotes the area of Γ.

Corollary 1.2. Let Hbe a 3-dimensional Cartan-Hadamard manifold with curvature

K≤a≤0, and Γ⊂ H be a C2closed strictly convex surface. Then

(1) G(Γ) ≥4π−a|Γ|,

with equality only if K≡aon the convex domain bounded by Γ.

Proof. By Gauss’ equation det(IIp) = KΓ(p)−K(TpΓ) for all p∈Γ, where KΓis the

intrinsic curvature of Γ, and TpΓis the tangent plane of Γat p. Since Γis simply

connected, RΓKΓ= 4πby Gauss-Bonnet theorem. Thus

(2) G(Γ) = 4π−Zp∈Γ

K(TpΓ) ≥4π−a|Γ|.

If equality holds in (1), then it also holds in (2), which forces K≡aon tangent planes of

Γ. Theorem 1.1, applied to the convex domain bounded by Γ, completes the proof. 

For a= 0, the last result is stated in [3, p. 66] and follows from [15, Thm. 2].

Gromov’s approach to the rigidity theorems mentioned above [3, Sec. 5], which are

further developed in [14,15], was based on extension of isometric embeddings in locally

symmetric spaces. In most of these results the rank of the space is required to be bigger

than 1, which precludes negative upper bounds for curvature. The arguments of Greene

and Wu [9] on the other hand involve volume comparison theory, which applies readily

to various curvature bounds [9, p. 734]; see Seshadri [17]. Here we develop a diﬀerent

approach via recent work on total curvature of Riemannian hypersurfaces [7,8], which

also yields some results for the dual case K≥a≥0; see Note 3.1. Generalizing (1) to

dimensions n > 3is an important open problem with applications to the isoperimetric

inequality; see [7] for more references and background in this area.

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RIGIDITYOFNONPOSITIVELYCURVEDMANIFOLDSWITHCONVEXBOUNDARYMOHAMMADGHOMIANDJOELSPRUCKAbstract.WeshowthatacompactRiemannian3-manifoldMwithstrictlyconvexsimplyconnectedboundaryandsectionalcurvatureKa0isisometrictoacon-vexdomaininacompletesimplyconnectedspaceofconstantcurvaturea,providedthatKaonplanest...

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RIGIDITY OF NONPOSITIVELY CURVED MANIFOLDS WITH CONVEX BOUNDARY MOHAMMAD GHOMI AND JOEL SPRUCK

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