A Survey on the Ricci flow on Singular Spaces
aa r X i v : . [ m a t h . DG ] J a n A SURVEY ON THE RICCI FLOW ON SINGULAR SPACES
KLAUS KR ¨ONCKE AND BORIS VERTMANA bstract . In this survey we provide an overview of our recent results con-cerning Ricci de Turck flow on spaces with isolated conical singularities. Thecrucial characteristic of the flow is that it preserves the conical singularity. Un-der certain conditions, Ricci flat metrics with isolated conical singularities arestable and positive scalar curvature is preserved under the flow. We also dis-cuss the relation to Perelman’s entropies in the singular setting, and outlineopen questions and future reseach directions. C ontents1 . Introduction and geometric preliminaries . Existence of the singular Ricci de Turck flow . Stability of the singular Ricci de Turck flow . Perelman’s entropies on singular spaces . Positive scalar curvature along singular Ricci de Turck flow . Open questions and further research directions Appendix A. Sobolev and H ¨older spaces References . I ntroduction and geometric preliminaries
Geometric flows, among them most notably the Ricci flow, provide a power-ful tool to attack classification problems in differential geometry and constructRiemannian metrics with prescribed curvature conditions. The interest in thisresearch area only grew since the Ricci flow was used decisively in the Perel-man’s proof of Thurston’s geometrization and the Poincare conjectures.The present article summarizes recent results of a continuation of a researchprogram on the Ricci flow in the setting of singular spaces, obtained in thepapers [V er16 , K r V e19a , K r V e19b ]. The two-dimensional Ricci flow reduces Date : January , . Mathematics Subject Classification.
Primary C ; Secondary C ; J . Key words and phrases.
Ricci flow, stability, integrability, conical singularities.Partial support by DFG Priority Programme ”Geometry at Infinity”. KLAUS KR ¨ONCKE AND BORIS VERTMAN to a scalar equation and has been studied on surfaces with conical singularitiesby Mazzeo, Rubinstein and Sesum in [MRS ] and Yin [Y in10 ]. The Ricci flowin two dimensions is equivalent to the Yamabe flow, which has been studiedin general dimension on spaces with edge singularities by Bahuaud and thesecond named author in [B a V e14 ] and [B a V e17 ].In the setting of K¨ahler manifolds, K¨ahler-Ricci flow reduces to a scalarMonge Ampere equation and has been studied in case of edge singularities inconnection to the recent resolution of the Calabi-Yau conjecture on Fano man-ifolds by Donaldson [D on11 ] and Tian [T ia12 ], see also Jeffres, Mazzeo andRubinstein [JMR ]. K¨ahler-Ricci flow in case of isolated conical singularitiesis geometrically, though not analytically, more intricate than edge singularitiesand has been addressed by Chen and Wang [C h W a15 ], Wang [W an ], as wellas Liu and Zhang [L i Z h14 ].We should point out that in the singular setting, Ricci flow loses its unique-ness and need not preserve the given singularity structure. In fact, Giesen andTopping [G i T o10 , G i T o11 ] constructed a solution to the Ricci flow on surfaceswith singularities, which becomes instantaneously complete. Alternatively, Si-mon [S im13 ] constructed Ricci flow in dimension two and three that smoothensout the singularity. Acknowledgements:
The authors thank the Geometry at Infinity Priority pro-gram of the German Research Foundation DFG as well as the Australian Math-ematical society for its financial support and for providing a platform for jointresearch. . . Isolated conical singularities.
Let us first put up an illustration of a com-pact Riemannian manifold with an isolated conical singularity in Figure , andthen proceed with a precise definition.F igure 1 . A compact manifold with an isolated conical singular-ity. SURVEY ON THE RICCI FLOW ON SINGULAR SPACES Definition . . Consider a compact smooth manifold M with boundary ∂M = F andopen interior denoted by M . Let C ( F ) be a tubular neighborhood of the boundary,with open interior C ( F ) = (
0, 1 ) x × F , where x is a defining function of the boundary.Consider a smooth Riemannian metric g F on the boundary F with n = dim F . Anincomplete Riemannian metric g on M with an isolated conical singularity is thendefined to be smooth away from the boundary and g ↾ C ( F ) = dx + x g F + h, where the higher order term h has the following asymptotics at x = . Let g = dx + x g F denote the exact conical part of the metric g over C ( F ) and ∇ g the correspondingLevi Civita connection. Then we require that for some γ > 0 and all integer k ∈ N the pointwise norm | x k ∇ kg h | g = O ( x γ ) , x → ( . ) Remark . . We emphasize here that we do not assume that the higher orderterm h is smooth up to x = and do not restrict the order γ > 0 to be integer. Inthat sense the notion of conical singularities in the present discussion is moregeneral than the classical notion of conical singularities where h is usuallyassumed to be smooth up to x = with γ = . This minor generalizationis necessary, since the Ricci de Turck flow, which will be introduced below,preserves a conical singularity only up to a higher order term h as above.We call ( M, g ) a compact space with an isolated conical singularity, or a con-ical manifold and g a conical metric. The definition naturally extends to conicalmanifolds with finitely many isolated conical singularities. Since the analyticarguments are local in nature, we may assume without loss of generality that M has a single conical singularity only.In the present discussion we study compact conical Ricci-flat manifolds ( M, g ) .There are various examples for such spaces. Consider a Ricci-flat smooth com-pact manifold X , e.g. a Calabi-Yau manifold or flat torus, with a discrete group G acting by isometries, which is not necessarily acting strictly discontinuousand admits finitely many fixed points. The interior of its quotient X/G definesa compact manifold, an orbifold, with isolated conical singularities. There ex-ist also examples of compact Ricci-flat manifolds with non-orbifold isolatedconical singularities, constructed by Hein and Sun [H e S u16 ].We now recall elements of b-calculus by Melrose [M el93 , M el92 ]. We chooselocal coordinates ( x, z ) on the conical neighborhood C ( F ) , where x is the defin-ing function of the boundary, n = dim F and ( z ) = ( z , . . . , z n ) are local co-ordinates on F . We consider the Lie algebra of b-vector fields V b , which bydefinition are smooth in the interior M and tangent to the boundary F . In local KLAUS KR ¨ONCKE AND BORIS VERTMAN coordinates ( x, z ) , b-vector fields V b are locally generated by (cid:12) x ∂∂x , ∂ z = (cid:18) ∂∂z , . . . , ∂∂z n (cid:19) (cid:13) , with coefficients being smooth on M . The b-vector fields form a spanning set ofsection for the b-tangent bundle b TM , i.e. V b = C ∞ ( M, b TM ) . The b-cotangentbundle b T ∗ M is generated locally by the following one-forms (cid:12) dxx , dz , . . . , dz n (cid:13) . ( . )These differential forms are singular in the usual sense, but smooth as sectionsof the b-cotangent bundle b T ∗ M . We extend x : C ( F ) → (
0, 1 ) smoothly to anon-vanishing function on M and define the incomplete b-tangent space ib TM by the requirement C ∞ ( M, ib TM ) = x − C ∞ ( M, b TM ) . The dual incomplete b-cotangent bundle ib T ∗ M is related to its complete counterpart by C ∞ ( M, ib T ∗ M ) = xC ∞ ( M, b T ∗ M ) , ( . )with the spanning sections given locally over C ( F ) by { dx, xdz , . . . , xdz n } . ( . )With respect to the notation we just introduced, the conical metric g in Defini-tion . is a smooth section of the vector bundle of the symmetric -tensors ofthe incomplete b-cotangent bundle ib T ∗ M , i.e. g ∈ C ∞ ( Sym ( ib T ∗ M )) . . . Ricci de Turck flow and the Lichnerowicz Laplacian.
The Ricci flow is anevolution equation for metrics which reads as ∂ t g ( t ) = − Ric ( g ( t )) . ( . )Due to diffemorphism invariance of the Ricci tensor, this evolution equationfails to be strongly parabolic. One overcomes this problem by adding an addi-tional term to the equation which brakes the diffeomorphism invariance. Forthis reason, one defines the Ricci de Turck flow ∂ t g ( t ) = − Ric ( g ( t )) + L W ( g ( t ) , e g ) g ( t ) ( . )where W ( t ) is the de Turck vector field defined in terms of the Christoffelsymbols for the metrics g ( t ) and a reference metric e gW ( g, e g ) k = g ij (cid:0) Γ kij ( g ) − Γ kij ( e g ) (cid:1) . ( . )This flow is equivalent to the Ricci flow via diffeomorphisms. The linearizationof the right hand side is given by dds − Ric ( e g + sh ) + L W ( e g + sh, e g ) ( e g + sh ) | s = = − ∆ L h, ( . ) SURVEY ON THE RICCI FLOW ON SINGULAR SPACES where ∆ L is an elliptic operator which is known as the Lichnerowicz Laplacian.Thus, the Ricci de Turck flow is parabolic in the strict sense and, at least in thesmooth compact setting, standard existens theorems guarantee well-posednessfor its initial value problem. Therefore from the analytical perspective, theRicci de Turck flow is much easier to handle than the standard Ricci flow.As it appears in its linearization, the Lichnerowicz Laplacian and its spectralproperties will be fundamental for considerations in this article. . E xistence of the singular R icci de T urck flow We shall present here the short time existence result for the singular Ricci deTurck flow obtained by the second author in [V er16 ] in a simple way, whichis sufficient for the purpose of the present discussion. We consider a compactconical manifold ( M, g ) . We study the Ricci de Turck flow with g as the initialmetric While the reference metric e g is usually taken as the initial metric g , incase of e g being Ricci flat, the initial metric g can be chosen as a sufficientlysmall perturbation of e g .To get shorttime existence for the Ricci de Turck flow in the conical setting,we need two more conditions. The first one is a condition where we requiresubquadratic blowup of the Ricci and scalar curvature close to the singularpoint. The second is a spectral condition on the Lichnerowicz Laplacian whichwe will explain in the following subsection. . . Tangential stability.
Let ( M, h ) be a compact conical Ricci-flat manifold.We write S := Sym ( ib T ∗ M ) . The Lichnerowicz Laplacian ∆ L : C ∞ ( M, S ) → C ∞ ( M, S ) is a differential operator of second order, that can be written in localcoordinates near the conical singularity as follows. We choose local coordinates ( x, z ) over the singular neighborhood C ( F ) = (
0, 1 ) x × F . In the previous paper[V er16 ] we have introduced a decomposition of compactly supported smoothsections C ∞ ( C ( F ) , S ↾ C ( F )) C ∞ ( C ( F ) , S ↾ C ( F )) → C ∞ ((
0, 1 ) , C ∞ ( F ) × Ω ( F ) × Sym ( T ∗ F )) ,ω → ( ω ( ∂ x , ∂ x ) , ω ( ∂ x , · ) , ω ( · , · )) , ( . )where Ω ( F ) denotes differential -forms on F . Under such a decomposition,the Lichnerowicz Laplace operator ∆ L associated to the singular Riemannianmetric g attains the following form over C ( F ) ∆ L = − ∂ ∂x − nx ∂∂x + (cid:3) L x + O , ( . )where (cid:3) L is a differential operator on C ∞ ( F ) × Ω ( F ) × Sym ( T ∗ F ) and the higherorder term O ∈ x − V is a second order differential operator with one orderhigher asymptotic behaviour at x = . KLAUS KR ¨ONCKE AND BORIS VERTMAN
Definition . . Let ( F n , g F ) be a closed Einstein manifold with Einstein constant ( n − ) . Then ( F n , g F ) is called (strictly) tangentially stable if the tangential operatorof the Lichnerowicz Laplacian on its cone restricted to tracefree tensors is non-negative(resp. strictly positive). . . The existence result.
The conditions for shorttime existence of the Riccide Turck flow are subsummarised under the notion of admissible metrics. Inorder to improve readability of the article, we put the definitions of the functionspaces to an appendix.
Definition . . Let ( M, g ) be a compact conical manifold. Then the conical metric g is said to be admissible , if it satisfies the following assumptions for γ > 0 as in ( . ) , some k ∈ N and α ∈ (
0, 1 ) . ( ) The cross section ( F, g F ) is assumed to be tangentially stable. ( ) The Ricci curvature has subquadratic growth at the singularitiy, i.e.
Ric ( g ) = O ( x − + γ ) as x → . More precisely, let scal ( g ) denote the scalar curvatureof g and Ric ◦ ( g ) the trace-free part of the Ricci curvature tensor. Then weassume scal ( g ) ∈ x − + γ C k + ie ( M, S ) , Ric ◦ ( g ) ∈ C k + ie ( M, S ) − + γ . ( . )( ) For any X , . . . , X ∈ C ∞ ( M, ib TM ) we have for the curvature (
0, 4 ) -tensor Rm ( g )( X , X , X , X ) ∈ x − C k + ie ( M ) . The main result of [V er16 , Theorem . ], see also [K r V e19a , Theorem . ], isthen the following theorem. Theorem . . Let ( M, g ) be a conical manifold with an admissible metric g . Let thereference metric e g be either equal to g or an admissible conical Ricci flat metric, inwhich case g is assumed to be a sufficiently small perturbation of e g in H k + ( M, S ) .Then there exists some T > 0 , such that the Ricci de Turck flow ( . ) with referencemetric e g , starting at g admits a solution g ( t ) , t ∈ [
0, T ] , which is an admissibleperturbation of g , i.e. g ( t ) ∈ H k + ′ ( M, S ) for each t , all k ∈ N and some γ ′ ∈ (
0, γ ) sufficiently small.Remark . . A precise definition of the space H k + ′ ( M, S ) is given in the ap-pendix. Let us give a brief description here. Decompose a symmetric -tensor h into its pure trace part and its tracefree part as h = tr h · g + h . Then If ( M, g ) is a conical manifold satisfying condition ( ) in Definition . , then the crosssection ( F, g F ) of the cone is automatically Einstein with Einstein constant ( n − ) . In view of Definition . ( . ) the condition ( ) is satisfied if the leading exact part g = dx + x g F of the conical metric g with g ↾ C ( F ) = g + h is Ricci flat, and the higher orderterm h not only satisfies ( . ), but in particular is an element of C k + ie ( M, S ) γ . SURVEY ON THE RICCI FLOW ON SINGULAR SPACES h ∈ H k + ′ ( M, S ) means that as we approach the singularity, ∇ l h = O ( x − l + γ ′ ) for l ∈ {
0, . . . , k + } , ∇ l tr h = O ( x − l + γ ′ ) for l ∈ {
1, . . . , k + } but tr h = O ( ) .The last condition distinguishes H k + ′ from a standard weighted H ¨older spaceand ensures that multiples of the metric are also contained in this space.This result is obtained as a consequence of a careful microlocal analysis ofthe heat kernel for the Lichnerowicz Laplacian. The heat kernel asymptotics isthen used. to establish mapping properties of the heat operator on the H ¨olderspaces H k + . It is here, that tangential stability enters in order to obtain thesemapping properties. Short time existence of the Ricci de Turck flow in thesespaces is then a consequence of a fixed point argument, which requires themetric to be admissible in the sense above to go through.Let us now explain in what sense the flow preserves the conical singularity.Given an admissible perturbation g of the conical metric g , the pointwisetrace of g with respect to g , denoted as tr g g is by definition of admissibilityan element of the H ¨older space C k ,α ie ( M, S ) bγ , restricting at x = to a constantfunction ( tr g g )( ) = u > 0 . Setting e x := √ u · x , the admissible perturbation g = g + h attains the form g = d e x + e x g F + e h, where | e h | g = O ( x γ ) as x → . Note that the leading part of the admissibleperturbation g near the conical singularity differs from the leading part of theadmissible metric g only by scaling. . . Characterizing tangential stability.
A crucial part of our paper [K r V e19a ]is devoted to a detailed discussion of the tangential stability. Namely, we provethe following general characterization. Theorem . . Let ( F, g F ) , n ≥ be a compact Einstein manifold with constant ( n − ) .We write ∆ E for its Einstein operator, and denote the Laplace Beltrami operator by ∆ .Then ( F, g F ) is tangentially stable if and only if Spec ( ∆ E | TT ) ≥ and Spec ( ∆ ) \ { } ∩ ( n, 2 ( n + )) = ∅ . Similarly, ( M, g ) is strictly tangentially stable if and only if Spec ( ∆ E | TT ) > 0 and Spec ( ∆ ) \ { } ∩ [ n, 2 ( n + )] = ∅ . Establishing this result amounts a careful anaylsis of the Lichnerowicz Lapla-cian. This analysis heavily relies on a decomposition of symmetric two tensorsestablished by the first author in [K r ¨ o17 ] to understand the spectrum of ∆ L onRicci-flat cones.Any spherical space form is tangentially stable because the LichnerowiczLaplacian of its cone is the rough Laplacian since the cone is flat. However, thespaces S n and RP n are not strictly tangentially stable since ( n + ) ∈ Spec ( ∆ ) in both cases. This property may also hold for other spherical space forms. In KLAUS KR ¨ONCKE AND BORIS VERTMAN the following theorem, we use Theorem . and eigenvalue computations in[C a H e15 ] to characerize tangential stability of symmetric spaces. Theorem . . Let ( F n , g F ) , n ≥ be a closed Einstein manifold with constant ( n − ) ,which is a symmetric space of compact type. If it is a simple Lie group G , it is strictlytangentially stable if G is one of the following spaces: Spin ( p ) ( p ≥
6, p = ) , E , E , E , F . ( . ) If the cross section is a rank- symmetric space of compact type G/K , ( M, g ) is strictlytangentially stable if G is one of the following real Grasmannians SO ( + + ) SO ( + ) × SO ( ) ( p ≥
2, q ≥ ) , SO ( ) SO ( ) × SO ( ) , SO ( ) SO ( p ) × SO ( p ) ( p ≥ ) , SO ( + ) SO ( p + ) × SO ( p ) ( p ≥ ) SO ( ) SO ( − q ) × SO ( q ) ( p − ≥ q ≥ ) , ( . ) or one of the following spaces: SU ( ) / SO ( p ) ( n ≥ ) , E / [ Sp ( ) / { ± I } ] , E / SU ( ) · SU ( ) , E / [ SU ( ) / { ± I } ] , E / SO ( ) · SU ( ) , E / SO ( ) , E / E · SU ( ) , F /Sp ( ) · SU ( ) . ( . ) . S tability of the singular R icci de T urck flow Our main result in [K r V e19a ] establishes long time existence and conver-gence of the Ricci de Turck flow for sufficiently small perturbations of conicalRicci-flat metrics, assuming linear and tangential stability and integrability.More precisely we consider a compact conical Ricci-flat manifold ( M, h ) and g a sufficiently small perturbation of h , not necessarily Ricci-flat. We studythe Ricci de Turck flow with h as the reference metric, and g as the initialmetric ∂ t g ( t ) = − Ric ( g ( t )) + L W ( t ) g ( t ) , g ( ) = g , ( . )where W ( t ) is W ( t ) k = g ( t ) ij (cid:0) Γ kij ( g ( t )) − Γ kij ( h ) (cid:1) . ( . ) Definition . . We say that ( M, h ) is linearly stable if the the Lichnerowicz Lapla-cian ∆ L with domain C ∞ ( M, S ) is non-negative. Definition . . We say that ( M, h ) is integrable if for some γ > 0 there exists asmooth finite-dimensional manifold F ⊂ H k,αγ ( M, S ) such that ( ) T h F = ker ∆ L,h ⊂ H k,αγ ( M, S ) , ( ) all Riemannian metrics h ∈ F are Ricci-flat. SURVEY ON THE RICCI FLOW ON SINGULAR SPACES Our main result is as follows.
Theorem . . Consider a compact conical Ricci-flat manifold ( M, h ) . Assume that ( M, h ) satisfies the following three additional assumptions (i) ( M, h ) is tangentially stable in the sense of Definition . , (ii) ( M, h ) is linearly stable in the sense of Definition . , (iii) ( M, h ) is integrable in the sense of Definition . .If h is not strictly tangentially stable, we assume in addition that the singularities areorbifold singularities. Then for sufficiently small perturbations g of h , there exists aRicci de Turck flow, with a change of reference metric at discrete times, starting at g and converging to a conical Ricci-flat metric h ∗ as t → ∞ .Remark . . If ( M, h ) is a smooth compact manifold, tantential stability is al-ways satisfied. In this case, the statement coincides with the stability results ofcompact smooth Ricci-flat manifolds obtained in [S es06 ].In contrast to the smooth compact case, we can not work with a priori esti-mates because the curvature is unbounded. Instead, the mapping propertiesof the heat kernel of the Lichnerowicz Laplacian on the spaces H k,αγ ( M, S ) playa pivotal role in our proof.We also study examples of compact conical manifolds where the integrabilitycondition is satisfied. This includes flat spaces with orbifold singularities aswell as K¨ahler manifolds. More precisely we establish the following results. Proposition . . Let ( M, h ) be a flat manifold with an orbifold singularity. Then itis linearly stable and integrable. The proof of this result is quite simple. Linear stability follows from the ab-sence of curvature and integrability follows from construction the submanifoldexplicitly as an affine space over h modelled over the space of parallel tensors. Theorem . . Let ( M, h ) be a Ricci-flat K¨ahler manifold where the cross sectionis either strictly tangentially stable or a space form. Then h is linearly stable andintegrable. This result has been obtained in the smooth compact case in the eighties(see e.g. [T ia87 ]). The result in our setting follows from carefully adoptingtechniques from the compact case and using the analysis of weighted Sobolevspaces. . P erelman ’ s entropies on singular spaces In this section, we review the results obtained in [K r V e19b ] on Perelman’sentropies on compact conical manifolds. From now on, let ( M m , g ) be a com-pact conical manifold and n = m − . We first introduce the three entropies ofinterest. KLAUS KR ¨ONCKE AND BORIS VERTMAN . . The λ -functional. The λ -functional is then defined as λ ( g ) = inf (cid:12)Z M ( scal ( g ) ω + | ∇ ω | ) dV g | ω ∈ H ( M ) , ω > 0, Z M ω dV g = (cid:13) . The corresponding Euler-Lagrange equation is g ω g + scal ( g ) ω g = λ ( g ) ω g , ( . ) . . The Ricci shrinker entropy.
Consider the functional W − ( g, f, τ ) W − ( g, ω, τ ) := ( ) m/2 Z M [ τ ( scal ( g ) · ω + | ∇ ω | ) − ln ω − m ω ] dV g . The Ricci shrinker entropy is then defined by µ − ( g, τ ) = inf (cid:12) W − ( g, ω, τ ) | ω ∈ H ( M ) , ω > 0, 1 ( ) m/2 Z M ω dV g = (cid:13) and the corresponding Euler Lagrange equation is τ (− g ω g − scal ( g ) ω g ) + log ( ω g ) ω g +( m + ν − ( g, τ )) ω g = It can be shown exactly as in [CCG+ , Corollary . ], that if λ ( g ) > 0 , thereal number ν − ( g ) = inf { µ − ( g, τ ) | τ > 0 } exists and is attained by a parameter τ g and a minimizing function ω g . . . The Ricci expander entropy.
Consider the functional W + ( g, f, τ ) W + ( g, ω, τ ) := ( ) m/2 Z M [ τ ( scal ( g ) · ω + | ∇ ω | ) + ln ω + m ω ] dV g . The expander entropy is then defined by µ + ( g, τ ) = inf (cid:12) W + ( g, ω, τ ) | ω ∈ H ( M ) , ω > 0, 1 ( ) m/2 Z M ω dV g = (cid:13) and the corresponding Euler Lagrange equation is τ (− g ω g − scal ( g ) ω g ) − log ( ω g ) ω g +(− m + ν + ( g, τ )) ω g = ( . )It is now shown exactly as in [FIN , p. ], that if λ ( g ) < 0 , the real number ν + ( g ) = sup { µ + ( g, τ ) | τ > 0 } exists and is attained by a parameter τ g and aminimizing function ω g .These functionals are defined almost exactly as in the smooth compact set-ting. The only difference is that one minimizes in the space H ( M ) instead of C ∞ ( M ) . However the weighted Sobolev space H ( M ) is exactly the right spaceas it covers the blowup of the scalar curvature at the singular points.The functionals λ and µ − were already studied by Dai and Wang in thepapers [D a W a18 , D a W a17 ] in the setting of compact conical manifolds. Theyshowed that they are well-defined and posess minimizers provided that the SURVEY ON THE RICCI FLOW ON SINGULAR SPACES scalar curvature of the cross section satisfies scal ( g F ) > m − . The minimizersare satisfying for any ε > 0 the asymptotics ω g ( x ) = o (cid:16) x − m − − ε (cid:17) as x → By the same methods, we obtained such analogous results for µ + under thesame assumptions [K r V e19b , Theorem . ].However, these results can be massively improved by restriction the assump-tions on the scalar curvature of the cross section. Theorem . . Let ( M m , g ) be a compact conical Riemannian manifold. Let n = m − and ( F n , g F ) be the cross section of the conical part of the metric g and assume that scal ( g F ) = n ( n − ) . Let ω g be a minimizer in the definition of the λ -functional,shrinker or the expander entropy. Then there exists an γ > 0 such that ω g admits apartial asymptotic expansion ω g ( x, z ) = const + O ( x γ ) , as x → and moreover for k ∈ N , | ∇ kg ω g | g ( x, z ) = O ( x γ − k ) , as x → This result is proved by writing the minimizers in terms of the heat operator,which allows us to use its mapping properties (Schauder estimates) as alreadyin the proof of the short time existence of the Ricci de Turck flow. Mappingproperties of the heat operator allow us to improve the asymptotics of the min-imizers incrementally and the statement is obtained by an iteration argument.This improvement allows us to study the relation to the Ricci solitons. Riccisolitions are Riemannian metrics g such that for its Ricci curvature Ric ( g ) , somevector field X , the Lie-derivative L X and a positive constant c > 0 , the followingequations are satisfiedRic ( g ) + L X g = (steady Ricci soliton) , Ric ( g ) + L X g = c g (shrinking Ricci soliton) . ( . )Any steady Ricci soliton is up to a diffeomorphism a constant solution to theRicci flow ∂ t g ( t ) = − Ric ( g ( t )) , g ( ) = g. ( . )Any shrinking Ricci soliton is up to a diffeomorphism a constant solution tothe normalized Ricci flow ∂ t g ( t ) = − Ric ( g ( t )) +
2c g ( t ) , g ( ) = g. ( . )Recall also that a Ricci soliton is called gradient if X = ∇ f for some function f : M → R . KLAUS KR ¨ONCKE AND BORIS VERTMAN
Using improved asymptotics we prove the following results, generalizingwell known theorems in the compact smooth setting.
Theorem . . Let ( M m , g ) be a compact conical Riemannian manifold. Let n = m − and ( F n , g F ) be the cross section of the conical part of the metric g . Then the followingstatements hold. (i) Suppose that m ≥ and scal ( g F ) = n ( n − ) . Then, if ( M, g ) is a Ricci soliton,it is gradient. Moreover, if ( M, g ) is steady or expanding, it is Einstein. (ii) In dimension m = , the assertions of part (i) hold if Ric ( g F ) = ( n − ) g F . In addition, we prove monotonicity of the entropies along the singular Riccide Turck flow.
Theorem . . Let ( M, g ) be a compact conical Riemannian manifold of dimension dim M ≥ and a tangentially stable cross section. (i) Then the λ -functional is nondecreasing along the Ricci de Turck flow preservingconical singularities and constant only along Ricci flat metrics. (ii) Whenever defined, the shrinker and the expanding entropies are nondecreasingalong the (normalized) Ricci de Turck flow preserving conical singularities andconstant only along shrinking and expanding solitons, respectively. . P ositive scalar curvature along singular R icci de T urck flow It is well known that a Ricci flow on smooth compact manifolds preservesthe condition of positive scalar curvature. This is an easy consequence of themaximum principle applied to the evolution equation on the scalar curvature.Moreover, the strong maximum principle implies that in this setting a metricof nonnegative scalar curvature that is not Ricci-flat will evolve to a metric ofpositive scalar curvature immediately. In all these cases, the Ricci flow becomesextinct after finite time.In current work in progress [KMV ], we establish such results also forthe singular Ricci flow on compact conical manifolds. However, we need tostrengthen our tangential stability assumption (which guarantees shorttimeexistence for the singular Ricci flow) somewhat further in order to obtain suchresults. In fact we need to assume that (cid:3) L > n on the orthogonal complementof constant functions. In that case we can obtain existence of singular Ricci deTurck flow in H k,αγ ( M × [
0, T ] , S ) with γ > 1 . In this case, the de Turck vec-tor field is sufficiently regular (i.e. it goes to zero as we approach the conicalsingularity). This allows us to prove the desired statement. Theorem . . The singular Ricci de Turck flow preserves positivity of scalar curvaturealong the flow, provided that the de Turck vector field is sufficiently regular. Moreover,
SURVEY ON THE RICCI FLOW ON SINGULAR SPACES if the initial metric has nonnegative scalar curvature and is not Ricci-flat, the Ricci deTurck flow will become extinct after finite time. The stronger assumption on the tangential operator can be characterizedexplicitly in a similar manner as in Theorem . . For brevity, we only providea list of symmetric spaces of compact type that satisfy that assumption. Theorem . . Amongst the symmetric spaces of compact type, only E , E / [ SU ( ) / { ± I } ] , E / SO ( ) , E / E · SU ( ) satisfy the conditions (cid:3) L > n . . O pen questions and further research directions One obvious but intricate future research direction is clearly an extensionof the analysis to non-isolated cones, so-called wedges, and more generallystratified spaces with iterated cone-wedge singularities. Already the existenceof the various entropies in the edge setting is an open question.On the other side it is clearly imperative to weaken the conditions of (strict)tangential stability and integrability for more general applications. This mightrequire a setup of the Ricci de Turck flow in L p based Sobolev spaces insteadof H ¨older spaces as the authors have done till now. This approach would alsoallow us to study the flow of singular metric without a subquadratic blowupof the Ricci curvature.Another question is whether we can descend to the Ricci flow. This dependson whether the de Turck vector field points to or out of the singularity.A ppendix A. S obolev and
H ¨ older spaces
Let ∇ g denote the corresponding Levi Civita covariant derivative. Let theboundary defining function x : C ( F ) → (
0, 1 ) be extended smoothly to M ,nowhere vanishing on M . We consider the space L ( M ) of square-integrablescalar functions with respect to the volume form of g . We define for any s ∈ N and any δ ∈ R the weighted Sobolev space H sδ ( M ) as the closure of compactlysupported smooth functions C ∞ ( M ) under k u k H sδ := s X k = k x k − δ ∇ kg u k L . (A. )Note that L ( M, E ) = H ( M, E ) by construction. KLAUS KR ¨ONCKE AND BORIS VERTMAN
Remark A. . An equivalent norm on the weighted Sobolev space H sδ ( M ) can bedefined for any choice of local bases { X , . . . , X m } of V b as follows. We omit thesubscript g from the notation of the Levi Civita covariant derivative and write k u k H sδ = s X k = X ( j , ··· ,j k ) k x − δ (cid:16) ∇ X j1 ◦ · · · ◦ ∇ X jk (cid:17) u k L . (A. ) Definition A. . The H¨older space C α ie ( M × [
0, T ]) , α ∈ [
0, 1 ) , consists of functions u ( p, t ) that are continuous on M × [
0, T ] with finite α -th H¨older norm k u k α := k u k ∞ + sup (cid:18) | u ( p, t ) − u ( p ′ , t ′ ) | d M ( p, p ′ ) α + | t − t ′ | α2 (cid:19) < ∞ , (A. ) where the distance function d M ( p, p ′ ) between any two points p, p ′ ∈ M is definedwith respect to the conical metric g , and in terms of the local coordinates ( x, z ) in thesingular neighborhood C ( F ) given equivalently by d M (( x, y, z ) , ( x ′ , y ′ , z ′ )) = (cid:0) | x − x ′ | + ( x + x ′ ) | z − z ′ | (cid:1) . The supremum is taken over all ( p, p ′ , t ) ∈ M × [
0, T ] . We now extend the notion of H ¨older spaces to sections of the vector bundle S = Sym ( ib T ∗ M ) of symmetric -tensors. Definition A. . Denote the fibrewise inner product on S induced by the Riemannianmetric g , again by g . The H¨older space C α ie ( M × [
0, T ] , S ) consists of all sections ω of S which are continuous on M × [
0, T ] , such that for any local orthonormal frame { s j } of S , the scalar functions g ( ω, s j ) are C α ie ( M × [
0, T ]) .The α -th H¨older norm of ω is defined using a partition of unity { φ j } j ∈ J subordi-nate to a cover of local trivializations of S , with a local orthonormal frame { s jk } over supp ( φ j ) for each j ∈ J . We put k ω k ( φ,s ) α := X j ∈ J X k k g ( φ j ω, s jk ) k α . (A. )Norms corresponding to different choices of ( { φ j } , { s jk } ) are equivalent andwe may drop the upper index ( φ, s ) from notation. The supremum norm k ω k ∞ is defined similarly. All the constructions naturally extend to sectionsin the sub-bundles S and S , where the H ¨older spaces for S reduce to theusual spaces in Definition A. . Finiteness of the H ¨older norm k u k α in particular implies that u is continuous on the closure M up to the edge singularity, and the supremum may be taken over ( p, p ′ , t ) ∈ M × [
0, T ] .Moreover, as explained in [V er16 ] we can assume without loss of generality that the tuples ( p, p ′ ) are always taken from within the same coordinate patch of a given atlas. SURVEY ON THE RICCI FLOW ON SINGULAR SPACES We now turn to weighted and higher order H ¨older spaces. We extend theboundary defining function x : C ( F ) → (
0, 1 ) smoothly to a non-vanishingfunction on M . The weighted H ¨older spaces of higher order are now definedas follows. Definition A. . ( ) The weighted H¨older space for γ ∈ R is x γ C α ie ( M × [
0, T ] , S ) := { x γ ω | ω ∈ C α ie ( M × [
0, T ] , S ) } , with H ¨older norm k x γ ω k α,γ := k ω k α . ( ) The hybrid weighted H¨older space for γ ∈ R is C α ie ,γ ( M × [
0, T ] , S ) := x γ C α ie ( M × [
0, T ] , S ) ∩ x γ + α C ie ( M × [
0, T ] , S ) with H ¨older norm k ω k ′ α,γ := k x − γ ω k α + k x − γ − α ω k ∞ . ( ) The weighted higher order H¨older spaces, which specify regularity of solutionsunder application of the Levi Civita covariant derivative ∇ of g on symmetric -tensors and time differentiation are defined for any γ ∈ R and k ∈ N by C k ,α ie ( M × [
0, T ] , S ) γ = { ω ∈ C α ie ,γ | { ∇ j V b ◦ ( x ∂ t ) ℓ } ω ∈ C α ie ,γ for any j + ≤ k } , C k ,α ie ( M × [
0, T ] , S ) bγ = { u ∈ C α ie | { ∇ j V b ◦ ( x ∂ t ) ℓ } u ∈ x γ C α ie for any j + ≤ k } , where the upper index b in the second space indicates the fact that despite theweight γ , the solutions u ∈ C k ,α ie ( M × [
0, T ] , S ) bγ are only bounded, i.e. u ∈ C α ie .The corresponding H¨older norms are defined using local bases { X i } of V and D k := { ∇ X i1 ◦ · · · ◦ ∇ X ij ◦ ( x ∂ t ) ℓ | j + ≤ k } by k ω k k + α,γ = X j ∈ J X X ∈ D k k X ( φ j ω ) k ′ α,γ + k ω k ′ α,γ , on C + α ie ( M × [
0, T ] , S ) γ , k u k k + α,γ = X j ∈ J X X ∈ D k k X ( φ j u ) k α,γ + k u k α , on C + α ie ( M × [
0, T ] , S ) bγ . ( ) In case of γ = we just omit the lower weight index and write e.g. C k ,α ie ( M × [
0, T ] , S ) and C k ,α ie ( M × [
0, T ] , S ) b . The H ¨older norms for different choices of local bases { X , . . . , X m } of V b anddifferent choices of conical Riemannian metrics g are equivalent due to com-pactness of M and F .The vector bundle S decomposes into a direct sum of sub-bundles S = S ⊕ S , (A. )where the sub-bundle S = Sym ( ib T ∗ M ) is the space of trace-free (with respectto the fixed metric g ) symmetric -tensors, and S is the space of pure trace Differentiation is a priori understood in the distributional sense. KLAUS KR ¨ONCKE AND BORIS VERTMAN (with respect to the fixed metric g ) symmetric -tensors. The sub bundle S istrivial real vector bundle over M of rank .Definition A. extends ad verbatim to sections of S and S . Since the sub-bundle S is a trivial rank one real vector bundle, its sections correspond toscalar functions. In this case we may omit S from the notation and simplywrite e.g. C k ,α ie ( M × [
0, T ]) bγ . Remark A. . The higher order weighted H ¨older spaces in Definition A. differslightly from the corresponding spaces in [V er16 ] by the choice of admissiblederivatives. While in [V er16 ] we allow differentiation by any b-vector field V ∈ V b , here we employ only derivatives of the form ∇ V , V ∈ V b .Below we will simplify notation by introducing the following spaces. Definition A. . Let ( M, g ) be a compact conical manifold and assume that the conicalcross section ( F, g F ) is strictly tangentially stable. Then we define H k,αγ ( M × [
0, T ] , S ) := C k ,α ie ( M × [
0, T ] , S ) γ ⊕ C k ,α ie ( M × [
0, T ] , S ) bγ . If ( F, g F ) is tangentially stable but not strictly tangentially stable, we set instead H k,αγ ( M × [
0, T ] , S ) := C k ,α ie ( M × [
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