On the singularities of spaces with bounded Ricci curvature.

*(English)*Zbl 1030.53046Authors’ introduction: “Theorems on partial regularity and on uniqueness of tangent cones play an important role in geometric analysis, e.g. in the theory of minimal surfaces and harmonic maps [see e.g. L. Simon, Theorems on regularity and singularity of energy minimizing maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag (Basel, 1996; Zbl 0864.58015)]. Here, we prove such theorems in the context of Riemannian geometry. In particular, this provides a framework for studying the degeneration of Einstein metrics.

Specifically, we are concerned with connected Riemannian manifolds \(M^n\), whose Ricci curvature and volume have definite bounds and whose curvature tensor has a definite \(L_p\)-bound, \(1\leq p\leq 2\). We describe the structure of spaces which are weak limits (Gromov-Hausdorff limits) of sequences of such manifolds.

A main result of ours states that if \(\{M_i^n\}\) is a Gromov-Hausdorff convergent sequence of Einstein manifolds with uniform bounds on the Einstein constant, the volume of a unit ball and the \(L_2\)-norm curvature, then the limit \(Y^n\), is a smooth Einstein manifold off a closed subset \(S\), of Hausdorff codimension \(4\). Off a subset of \(S\) with \((n-4)\)-dimensional Hausdorff measure \(0\), all tangent cones are of the form \(\mathbb{R}^{n-4}\times \mathbb{R}^4/\Gamma\), where \(\Gamma\) is a finite group of orthogonal transformations which acts freely on \(\mathbb{R}^4\setminus\{0\}\).

Well-known examples of limit spaces satisfying our assumptions exhibit codimension \(4\) singularities which are actually of orbifold type. The examples arise from asymptotically locally Euclidean spaces and, in the compact case, from gluing constructions. Many such examples are constructed in the monograph by D. Joyce [Compact manifolds with special holonomy, Oxford University Press, Oxford (2000; Zbl 1027.53052)]. The case, \(n=4\), \(p=2\), was pioneered by M. T. Anderson [J. Am. Math. Soc. 2, 455-490 (1989; Zbl 0694.53045); Invent. Math. 102, 429-445 (1990; Zbl 0711.53028); Geom. Funct. Anal. 2, 29-89 (1992; Zbl 0768.53021); Proc. Int. Congress Math., Zürich 1994, Vol. 1, 443-452 (1995; Zbl 0840.53036)], S. Bando, A. Kasue and H. Nakajima [Invent. Math. 97, 313-349 (1989; Zbl 0682.53045)] and G. Tian [Invent. Math. 101, 101-172 (1990; Zbl 0716.32019)].

In higher dimensions, the exponent, \(p=2\), continues to play a distinguished role, particularly for Kähler-Einstein manifolds, or more generally, manifolds with special holonomy. In the Kähler-Einstein case, the \(L_2\)-norm of the curvature can be bounded in terms of a characteristic number depending on the first two Chern classes and the Kähler class. A similar relation holds for manifolds with special holonomy [see J. Cheeger and G. Tian, Anti-self-duality of curvature and Riemannian metrics with special holonomy, in preparation). However, for \(n>4\), the \(L_2\)-norm of the curvature is not a scale invariant quantity, and typically, the singular set of a limit space does not consists of isolated points. This makes the analysis substantially harder and new techniques are required.

The starting point is the structure theory for Gromov-Hausdorff limits of sequences of manifolds with a uniform lower Ricci curvature bound developed in the paper by J. Cheeger and T. H. Colding [J. Differ. Geom. 46, 406-480 (1997; Zbl 0902.53034)]. This theory puts strong restrictions on the kinds of singularities which might potentially appear. For instance, in our situation, to rule out singularities in codimension \(2\), it turns out to be enough to show that singularities of the form \(\mathbb{R}^{n-2}\times C(S_t^1)\) cannot arise as limits of sequences for which (near the singularity) the \(L_2\)-norm of the curvature tensor goes to zero. Here \(C(S_t^1)\) denotes the cone whose cross-section is a circle of circumference \(t\leq 2\pi\). A feeling for the proof of this fact can be gained from the following elementary \(2\)-dimensional example. Since \(n=2\), technical difficulties that are present in higher dimensions do not appear.

Consider a \(2\)-sphere \(Y^2\) with a single isolated conical singularity, e.g. the surface of an ice cream cone. Assume that \(Y^2\) is smooth, except at the cone tip. We ask if it is possible to replace \(Y^2\) by a smooth manifold \(M^2\) which coincides with \(Y^2\) except in a tiny neighborhood, \(U\), of the cone tip, and for which the total amount of curvature contained in \(U\) is very small. Although \(U\) might be too small to observe directly, we can conclude that the the proposed smoothing is impossible, by noting that metric on the resulting manifold would not obey the Gauss-Bonnet formula.

Specifically, we are concerned with connected Riemannian manifolds \(M^n\), whose Ricci curvature and volume have definite bounds and whose curvature tensor has a definite \(L_p\)-bound, \(1\leq p\leq 2\). We describe the structure of spaces which are weak limits (Gromov-Hausdorff limits) of sequences of such manifolds.

A main result of ours states that if \(\{M_i^n\}\) is a Gromov-Hausdorff convergent sequence of Einstein manifolds with uniform bounds on the Einstein constant, the volume of a unit ball and the \(L_2\)-norm curvature, then the limit \(Y^n\), is a smooth Einstein manifold off a closed subset \(S\), of Hausdorff codimension \(4\). Off a subset of \(S\) with \((n-4)\)-dimensional Hausdorff measure \(0\), all tangent cones are of the form \(\mathbb{R}^{n-4}\times \mathbb{R}^4/\Gamma\), where \(\Gamma\) is a finite group of orthogonal transformations which acts freely on \(\mathbb{R}^4\setminus\{0\}\).

Well-known examples of limit spaces satisfying our assumptions exhibit codimension \(4\) singularities which are actually of orbifold type. The examples arise from asymptotically locally Euclidean spaces and, in the compact case, from gluing constructions. Many such examples are constructed in the monograph by D. Joyce [Compact manifolds with special holonomy, Oxford University Press, Oxford (2000; Zbl 1027.53052)]. The case, \(n=4\), \(p=2\), was pioneered by M. T. Anderson [J. Am. Math. Soc. 2, 455-490 (1989; Zbl 0694.53045); Invent. Math. 102, 429-445 (1990; Zbl 0711.53028); Geom. Funct. Anal. 2, 29-89 (1992; Zbl 0768.53021); Proc. Int. Congress Math., Zürich 1994, Vol. 1, 443-452 (1995; Zbl 0840.53036)], S. Bando, A. Kasue and H. Nakajima [Invent. Math. 97, 313-349 (1989; Zbl 0682.53045)] and G. Tian [Invent. Math. 101, 101-172 (1990; Zbl 0716.32019)].

In higher dimensions, the exponent, \(p=2\), continues to play a distinguished role, particularly for Kähler-Einstein manifolds, or more generally, manifolds with special holonomy. In the Kähler-Einstein case, the \(L_2\)-norm of the curvature can be bounded in terms of a characteristic number depending on the first two Chern classes and the Kähler class. A similar relation holds for manifolds with special holonomy [see J. Cheeger and G. Tian, Anti-self-duality of curvature and Riemannian metrics with special holonomy, in preparation). However, for \(n>4\), the \(L_2\)-norm of the curvature is not a scale invariant quantity, and typically, the singular set of a limit space does not consists of isolated points. This makes the analysis substantially harder and new techniques are required.

The starting point is the structure theory for Gromov-Hausdorff limits of sequences of manifolds with a uniform lower Ricci curvature bound developed in the paper by J. Cheeger and T. H. Colding [J. Differ. Geom. 46, 406-480 (1997; Zbl 0902.53034)]. This theory puts strong restrictions on the kinds of singularities which might potentially appear. For instance, in our situation, to rule out singularities in codimension \(2\), it turns out to be enough to show that singularities of the form \(\mathbb{R}^{n-2}\times C(S_t^1)\) cannot arise as limits of sequences for which (near the singularity) the \(L_2\)-norm of the curvature tensor goes to zero. Here \(C(S_t^1)\) denotes the cone whose cross-section is a circle of circumference \(t\leq 2\pi\). A feeling for the proof of this fact can be gained from the following elementary \(2\)-dimensional example. Since \(n=2\), technical difficulties that are present in higher dimensions do not appear.

Consider a \(2\)-sphere \(Y^2\) with a single isolated conical singularity, e.g. the surface of an ice cream cone. Assume that \(Y^2\) is smooth, except at the cone tip. We ask if it is possible to replace \(Y^2\) by a smooth manifold \(M^2\) which coincides with \(Y^2\) except in a tiny neighborhood, \(U\), of the cone tip, and for which the total amount of curvature contained in \(U\) is very small. Although \(U\) might be too small to observe directly, we can conclude that the the proposed smoothing is impossible, by noting that metric on the resulting manifold would not obey the Gauss-Bonnet formula.

Reviewer: Mircea Craioveanu (Timişoara)

##### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |