Some Remarks on Gradient Ricci Solitons Alberto G. Setti Universit` - - PowerPoint PPT Presentation

Some Remarks on Gradient Ricci Solitons

Alberto G. Setti Universit` a dell’Insubria - Como, Italy alberto.setti@uninsubria.it Joint work with Stefano Pigola (Universita’ dell’Insubria), Marco Rigoli and Michele Rimoldi (Universita’ di Milano) Convegno Nazionale di Analisi Armonica: 15-19 Giugno 2009 .

Let M be a n-dimensional manifold without boundary. The Ricci flow is the degenerate parabolic system on metrics on M defined by ∂tg(t) = −2Ric(g(t)) where g(t) is a time dependent unknown Riemannian metric and Ric(g) is its Ricci curvature. The Ricci flow was introduced by Hamilton in the early 80’s and it has proven the key tool in the Hamilton-Perelmam solution of the Poincare’ conjecture (and of Thurston’s geometrization conjecture).

Recall that the Poincare’ conjecture stated that every homotopic 3- sphere is (homeomorphic to) a round 3-sphere. For n-spheres with n ≥ 5 this was proved by Smale in 1961 and for n = 4 it was proved by Friedman in 1982 (using different techniques). Since the case of the two sphere is easy only the case n = 3 remained open. Hamilton’s basic idea was that the Ricci flow, which is a sort of heat equation on metrics, should smooth out the metric, and produce an increasingly homogeneous metric (just like the heat flow on compact manifolds tends to a uniform heat distribution). For instance, using the Ricci flow Hamilton proved:

Theorem 1 (Hamilton, JDG 1982) Let (M, g) be a compact Rie- mannian manifold with positive Ricci curvature Ric (in the sense of quadratic forms). Then M is diffeomorphic to a spherical space form, that is the 3-sphere S3 or a quotient of it by a finite group of fixed point free isometries in the standard metric. So a possible way of proving the Poincare conjecture was to endow the manifold with a metric, let it evolve by the Ricci flow, and hopefully end up with a positive constant curvature metric (which in dimension 3 characterizes the sphere and its quotients).

The problem is that unlike the heat equation, the Ricci flow develops

• singularities. In order to continue the flow past the singularities, Hamil-

ton wanted to apply surgery, cut out the singularity, replace it with a a nice manifold, and let the new Ricci flow run. In order to carry out this program it was vital to have a sufficiently good knowledge of the kind

• f singularities that can occur, and to control both the Ricci flow and

the topology across the singularities. Hamilton was able to obtain preliminary results in this direction, but was not able to treat all the singularities that could occur, and so was stuck for a rather long time, until Grisha Perelman came along, and in a series of three rather dense and extremely hard and sometime sketchy papers pushed through, solved all the technical problems and proved the Poincare’ conjecture and the Geometrization Conjecture.

Ricci solitons and the Ricci flow Definition 1 Let (M, g) be a Riemannian manifold. A Ricci soliton structure on M is a smooth vector field X satisfying the soliton equation Ric + 1 2LXg = λg, (1) for some constant λ ∈ R. Here: Ric = Ricci curvature of M; LX = Lie derivative in the direction X (a sort of directional derivative in the direction of X for tensor fields, built using the local 1-parameter group of diffeomorhisms generated by X) The Ricci soliton is contractive if λ > 0; steady if λ = 0; expansive if λ < 0.

If X = ∇f for some smooth function f : M → R, we say that (M, g, ∇f) is a gradient Ricci soliton with potential f. In this situation, the soliton equation reads Ric + Hess (f) = λg. (2) Ricci solitons are a generalization of Einstein manifolds (which are ”triv- ial” Ricci solitons corresponding to f = const). They give rise to self similar solutions of the Ricci flow and arise as the blow up of some of the singularities of the Hamilton Ricci flow ∂g(t) = −2Ric(g(t))

If (M, g) is an Einstein manifolds, so that Ric = λg, then g(t) = (1 − 2λt)g is a solutions of the Ricci flow defined on (−∞, 1/2λ) if λ > 0, on R if λ = 0 and on (1/2λ, +∞) if λ < 0. Indeed, since Ric(αg) = Ric(g) for every α > 0 ∂g(t) = −2λg = −2Ric(g) = −2Ric(g(t)) More generally, if g is a Ricci soliton satisfying (1), we set σ(t) = 1−2λt and let ϕt be the one parameter family of diffeomorphisms induced by the time-dependent vector field Yt = σ−1(t)X, so that ∂tϕt(x) = Yt(ϕt(x)) then g(t) = σ(t)ϕ∗

t(g)

is a self similar solution of the Ricci flow equation.

We will focus our attention on geodesically complete, gradient Ricci

• solitons. Here are some typical examples.

Example 2 The standard Euclidean space (Rm, gcan, ∇f) with f (x) = 1 2A |x|2 + x, B + C, for arbitrary A ∈ R, B ∈ Rm and C ∈ R, is an example of gradient Ricci

• soliton. Note that f is the essentially unique solution of the equation

Hess(f) = Acan on Rm. This follows integrating on [0, |x|] the equation d2 ds2 (f (vs)) = A, with v ∈ Rm such that |v| = 1. In fact, a kind of converse also holds. A complete manifold supporting a smooth function whose Hessian is a non-zero multiple of the metric, is isometric to Rn (Tashiro, TAMS, 1965). Question: Are there non trivial Ricci soliton structures on Sm and Hm.

If we restrict to gradient Ricci solitons, the answer is negative: Proposition 3 Let (M, g) be a gradient Ricci soliton with potential f, where M Einstein with Ric = cg. Then, either f ≡ constant and the soliton is trivial, or c = 0. Corollary 4 The spaces Sm, Hm S2 × S2, H2 × H2, with their canoni- cal metrics and their quotients don’t admit non-trivial gradient Ricci solitons. Example 5 Let us consider

• R × Sm, dt2 + can
• and f (x) = At2+Bt+C,

for arbitrary A, B, C ∈ R. Then Hess (f) = 2Adt2 and Ric = (m − 1) can. It follows that λdt2 + λcan = Ric + Hess (f) = 2Adt2 + (m − 1) can, and so, if (m − 1) = λ = 2A, the cylinder is an example of contractive gradient Ricci soliton.

Example 6 Let us consider

• R × Hm, dt2 + can
• .

In the same manner as above we get an expanding gradient Ricci soliton with λ = − (m − 1). Observe that it is non compact. Example 7 The Riemannian product

• Rm × Nk, gRm + gNk, ∇f
• where (Nk, gNk) is any k-dimensional Einstein manifold with Ricci cur-

vature λ = 0, and f (x, p) : Rm × Nk → R is defined by f (x, p) = λ 2 |x|2

Rm + gRm(x, B) + C,

with C ∈ R and B ∈ Rm.

The main goal is

• describe the geometric properties of Ricci solitons in terms of natural

geometric quantities

• if possible, classify Ricci solitons

If (M, g, ∇f) is a gradient Ricci soliton, by definition Ric + Hess(f) = λg The LHS is the so called Baky-Emery Ricci curvature associated to the weighted manifold (M, g, e−fdV ), where dV is the usual Riemannian measure, and to the diffusion operator ∆fu = efdiv(e−f∇u) = ∆u − g(∇f, ∇u), (3) which is self-adjoint on L2(M, e−fdV ) .

Note: The assertion that (M, g, ∇f) is a gradient Ricci soliton amounts to saying that the Bakry -Emery Ricci tensor Ricf of the weighted Riemannian manifold (M, g, e−fdV ) is constant. Thus Ricf is the most natural geometric object associated to a gradient Ricci soliton. Problem: A control on Ricf is not enough to obtain the kind of ge-

• metric information that can be obtained from a lower bound on the

Ricci curvature.

• For instance:

If M is geodesically complete and Ric ≥ (m − 1)cg in the sense of quadratic forms then

• if c > 0 then the manifold is compact, diam(M) ≤ π/√c and M has

finite fundamental group (Myers Theorem);

• if c = 0, vol (B(xo, r) ≤ vol RmBRm(r);
• if c < 0, vol (B(xo, r) ≤ vol Hm

c BHm c (r), where Hm

c

is Hyperbolic space with curvature c.

These conclusions do not follow from the analogous assumptions on Ricf, for instance Ricf ≥ cg, with c > 0 does not imply compactness (but does imply that M has finite volume and finite fundamental group). Indeed, the Bakry-Emery Ricci tensor is the limit of a family of modified Ricci tensors defined by Ricf,q = Ricf − 1 q d f ⊗ d f q ∈ (0, ∞) which are more suitable to control the geometry of the weighted mani- fold (M, g, e−fdV ).

The equations: Proposition 8 Let (M, g, ∇f) be a gradient Ricci soliton with Ric + Hessf = λg. Then the following Bochner-Weitzenb¨

• ck type formulae

hold: (i) 1

2∆ |∇f|2 = |Hess(f)|2 − Ric (∇f, ∇f).

(ii) 1

2∆ |∇f|2 = |Hess(f)|2 + 1 2g(∇ |∇f|2 , ∇f) − λ |∇f|2.

In terms of the operator ∆f (ii) can be stated also in the following way 1 2∆f |∇f|2 = |Hess (f)|2 − λ |∇f|2 . (4)

Combining this with the Kato inequality |Hess (f)|2 ≥ |∇ |∇f||2 , (5) we deduce the next Corollary 9 Let (M, g, ∇f) be a gradient Ricci soliton. Then, |∇f| ∈ Liploc (M) satisfies |∇f| ∆ |∇f| ≥ −Ric (∇f, ∇f) (6) and |∇f| ∆f |∇f| ≥ −λ |∇f|2 , (7) weakly on M.

Some results in the compact settings: Theorem 10 Let (M, g, ∇f) be a compact steady or expanding gradient Ricci soliton. Then it is trivial.

• Proof. According to formula (ii) in Proposition 8,

1 2∆ |∇f|2 = |Hess(f)|2 + 1 2g(∇ |∇f|2 , ∇f) − λ |∇f|2 ≥ 1 2g(∇ |∇f|2 , ∇f) − λ |∇f|2 . By the strong maximum principle applied to u = |∇f|2 we deduce the |∇f|2 is constant. Since ∇f necessarily vanishes on M (M is compact and f is smooth) we conclude that |∇f| ≡ 0 and f is constant.

Consider now M non compact. We are going to prove triviality of expanding Ricci solitons under the assumption that |∇f| belongs to Lp(e−fdV ) for some 1 < p ≤ +∞. Indeed, the same conclusion holds when p = 1 under the additional requirement |∇f| (x) = O

• eδr(x)2

, as r (x) → +∞, for some constant δ > 0, where r (x) is the intrinsic distance from a fixed origin.

These results are based on the following volume estimate obtained by

• F. Morgan (Notices of the AMS, 2005) and G. Wei W.Wylie (arXiv:

0760.1120). Theorem 11 Let

• M, g, e−fdV
• be a geodesically complete weighted
• manifold. Suppose that

Ricf ≥ λg, (8) for some constant λ ∈ R. Then, having fixed R0 > 0, there are constants A, B, C > 0 such that, for every r ≥ R0, volf (Br) ≤ A + B

r

R0

e−λt2+Ctdt. (9)

Therefore, a consequence of general results due to Pigola Rigoli S. we deduce the following Theorem 12 Let (M, g, e−fdV ) be a weighted Riemannian manifolds satisfying Ricf = Ric + Hessf ≥ λg.

• If λ > 0 then the weighted manifold (M, g, e−fdV ) is parabolic for the
• perator ∆f, namely if u is bounded above and satisfies ∆fu ≥ 0,

then u is constant.

• For every λ the weak maximum principle at infinity holds for the
• perator ∆f, namely, for every u ∈ C2 bounded above, there exists

a sequence {xn} in M such that (i) u(xn) ≥ sup

M

u − 1 n, (ii) ∆fu(xn) ≤ 1 n.

Corollary 13 (PRS) Let (M, g, ∇f) be an expanding gradient Ricci soli-

• ton. If |∇f| is bounded then the Ricci soliton is trivial.
• Proof. According to (4)

1 2∆f|∇f|2 = |Hessf|2 − λ|∇f| ≥ −λ|∇f|2. (10) By the weak maximum principle there exists {xn} such that |∇f(xn)| ≥ sup

M

|∇f| − 1/n and ∆|∇f|2(xn) ≤ 1/n. Evaluating the above inequality along xn and passing to the limit we deduce that −λ sup

M

|∇f| ≤ 0 and f is constant. This kind of arguments yields also the following scalar curvature esti- mate which improves on previous results by P. Petersen and Wylie.

Theorem 14 [PRS] Let (M, g, ∇f) be a geodesically complete gradient Ricci soliton with scalar curvature S satisfying S∗ = infM S > −∞. If M is expanding then mλ ≤ S∗ ≤ 0; if M is contractive then 0 ≤ S∗ ≤ mλ, and S∗ < mλ unless the soliton is trivial and M is compact Einstein.

• Proof. A computation shows that

1 2∆fS = λS − |Ric|2 . and since |Ric|2 ≥ 1

mS2

1 2∆fS ≤ λS − 1 mS2. (11) By the weak maximum principle there exists {xn} such that ∆fS(xn) ≥ −1/n and S(xn) → S∗, and taking the liminf in (11) along {xn} shows that λS∗ − S2

∗ /m ≥ 0. Thus, if λ < 0, then mλ ≤ S∗ ≤ 0, while, if λ > 0,

then 0 ≤ S∗ ≤ mλ.

Assume now that S∗ = λm > 0. Then S ≥ S∗ = mλ and λS − 1

mS2 ≤ 0.

It follows from (11) that S > 0 satisfies ∆fS ≤ 0. Since in this case M is f-parabolic, S = S∗ = mλ is constant, and |Ric|2 = 1

• mS2. By the

equality case in the Cauchy-Schwarz inequality, we deduce that Ric = λg with λ > 0 and M is compact by Myers’ Theorem. By definition Ric + Hess (f) = λg. and therefore Hess(f) = 0, and in particular f is a harmonic function

• n M compact, and therefore it is constant.

The proof that a Ricci soliton (M, g, ∇f) is trivial if |∇f| ∈ Lp(e−fdV ) 1 < p < +∞) is a consequence of the following general Liouville Theorem (PRS) Theorem 15 (PRS) Let

• M, g, e−fdV
• be a geodesically complete weighte
• manifold. Assume that u ∈ Liploc (M) satisfy

u∆fu ≥ 0, weakly on M. (12) If, for some p > 1, 1

• ∂Br |u|p e−fdvolm−1

/ ∈ L1 (+∞) , (13) then u is constant. Note that (13) certainly holds if u ∈ Lp(e−fdV ). Recalling that u = |∇f| satisfies |∇f| ∆f |∇f| ≥ −λ |∇f|2 ≥ 0, weakly on M. an application of Theorem 15 gives that |∇f| is constant. Using this information into the above inequality we conclude that |∇f| = 0 and f is a constant function.