DAN KNOPF AND NATAŠA ŠEŠUM

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1 DYNAMIC INSTABILITY OF UNDER RICCI FLOW DAN KNOPF AND NATAŠA ŠEŠUM Abstract. The intent of this short note is to provie context for an an inepenent proof of the iscovery of Klaus Kröncke [Kro13] that complex projective space with its canonical Fubini Stuy metric is ynamically unstable uner Ricci flow in all complex imensions N. The unstable perturbation is not Kähler. This provies a counterexample to a well known conjecture wiely attribute to Hamilton. Moreover, it shows that the expecte stability of the subspace of Kähler metrics uner Ricci flow, another conjecture believe by several experts, nees to be interprete in a more nuance way than some may have expecte. Contents 1. Introuction 1. Perelman s entropy 3 3. Variational formulas 4 4. Proof of the main result 10 References 1 1. Introuction Ricci solitons are stationary solutions of the Ricci flow ynamical system on the space of Riemannian metrics, moulo iffeomorphism an scaling. As such, it is natural to investigate their stability. Several istinct but relate notions of stability appear in the literature. i) One says a soliton metric g 0 is ynamically stable if for every neighborhoo V of g 0 there exists a neighborhoo U V such that every moifie) Ricci flow solution originating in U remains in V an converges to a soliton in V. This is subtly ifferent from the classical efinition for continuoustime ynamical systems for two reasons. Unless a soliton is Ricci flat, one nees to moify the flow to make the soliton into a bona fie fixe point. Furthermore, stationary solutions frequently occur in families, so one typically has to eal with the presence of a center manifol.) ii) One says g 0 is variationally stable if the secon variation of an appropriate choice of Perelman s energy or entropy functionals is nonpositive at g 0. Note that the expaner entropy was efine in [FIN05].) iii) One says that g 0 is linearly stable if the linearization of Ricci flow at g 0 has nonpositive spectrum. Variational an linear stability are equivalent in the compact case, in the precise sense that the secon-variation operator N efine in 3) below 1

2 DAN KNOPF AND NATAŠA ŠEŠUM satisfies N = l 1/τ when acting on trace-free ivergence-free tensors, where l is the Lichnerowicz Laplacian. 1 Techniques to prove the implication iii) i) for flat metrics were first evelope by one of the authors an collaborators. [GIK0]. Generalizing these, the other author prove for Ricci-flat metrics that i) iii), an that the converse hols if g 0 is integrable [Ses06]. Haslhofer Müller remove the integrability assumption, replacing it with the slightly stronger hypothesis that g 0 is a local maximizer of Perelman s λ-functional [HM14]. This must be verifie irectly an oes not follow from ii), because these metrics are only weakly variationally stable. In any case, the implication i) ii) is clear for all compact solitons. There are many other applications of stability theory to Ricci flow, too numerous to survey thoroughly here. An intereste reaer may consult the following not complete!) list of examples, orere by publication year: [Ye93], [DWW05], [DWW07], [SSS08], [Kno09], [LY10], [SSS11], [Has1], [Wu13], [Bam14], an [WW16]. Kröncke extene the techniques of Haslhofer Müller [HM14] to stuy stability at general compact Einstein manifols. Our interest here is narrower. In his investigation, Kröncke iscovere Corollary 1.8) the unexpecte fact that complex projective space with its canonical Fubini Stuy metric is ynamically unstable. We note that Kröncke s work [Kro13] was accepte for publication after the original version of this paper was complete. Nonetheless, we believe his instability result eserves inepenent attention because of its consierable broa interest, for at least three reasons: a) The result negatively answers a well-known conjecture wiely attribute to Hamilton. See, e.g., the iscussion of his conjecture in the introuction to [CZ1], as well as Hamilton s own analysis of the behavior of CP in Section 10 of [Ham95]. b) The result shows that the conjecture of experts that the subspace of Kähler metrics is an attractor for Ricci flow involves more complicate ynamical behavior than some may have expecte. It is well known that, g FS ) is stable uner Kähler perturbations [TZ13]. An it is easy to see that its unstable perturbation is not Kähler. However, if a flow originating at this perturbation asymptotically approaches the subspace of Kähler metrics, monotonicity of Perelman s shrinker entropy ν implies that it cannot o so at the nearest Kähler caniate, g FS ). Rather, if it converges to a Kähler singularity moel, that metric must be sufficiently far away. c) Finally, in real imension n = 4, ynamic instability of CP shows that only the first four metrics in the list of conjecture stable singularity moels, S 4 > S 3 R > S R > L 1 > CP, orere by the central ensity Θ introuce in [CHI04], are caniates to be generic singularity moels. Here, L 1 enotes the blowown soliton iscovere in [FIK03]. At least in this imension, it is reasonable to conjecture that solutions originating at the unstable perturbation 1 There is a subtlety in this equivalence an in proving iii) i) that shoul be note. The linearization of Ricci flow is not strictly parabolic, because the flow is invariant uner the action of the infinite-imensional iffeomorphism group. So one either has to work orthogonally to the orbit of that group, or else fix a gauge, which converts the linearization into the Lichnerowicz Laplacian. Such a choice is necessary if one wishes to prove ynamic stability by stanar semigroup methos, but can create a finite-imensional unstable manifol within the orbit of the iffeomorphism group, especially in the presence of positive curvature. See, e.g., Lemma 7 an the remarks that follow in [Kno09].

3 DYNAMIC INSTABILITY OF UNDER RICCI FLOW 3 of CP become singular in finite time by crushing the istinguishe fiber CP 1 CP, an converge after parabolic rescaling to L 1. For further context, see the iscussion an relate results in [IKS17]. Because of these important consequences, we believe that it is valuable to prouce an inepenent proof of Kröncke s iscovery, using relate but istinct methos. That is the purpose of this short note, in which we reprove the following: Main Theorem. For all complex imensions N, complex projective space with its canonical Fubini Stuy metric,, g FS ), is not a local maximum of Perelman s shrinker entropy ν. Consequently, it is ynamically unstable uner Ricci flow. The unstable perturbation is conformal but not Kähler.. Perelman s entropy Perelman s entropy functional W, introuce in [Per0], is efine on a close Riemannian manifol M n, g) by { Wg, f, τ) = τ R + f ) } + f n) 4πτ) n e f V. M n Suitably minimizing W yiels the shrinker entropy, ν[g] = inf {Wg, f, τ): f C M n ), τ > 0, 4πτ) n/ } e f V = 1. Along a smooth variation gs) such that s gs) = h, Perelman showe that ν [ gs) ] = 4πτ) n/ 1 s M g τrc + f), h e f V. n From this, he obtaine the beautiful result that ν is nonecreasing along any compact solution of Ricci flow in fact, strictly increasing except on graient shrinking solitons normalize so that 1) Rc + f = 1 τ g. As originally observe in [CHI04] see [CZ1] for etails, along with a complete erivation of the strictly more complicate formula that hols at a nontrivial shrinking soliton if gs) is a smooth family of metrics on M n such that g = g0) is Einstein, then the secon variation of ν at s = 0 is given by ) where V = VolM n, g), an s ν [ gs) ] = τ V M n Nh), h V, 3) Nh) = 1 h + Rmh, ) + iv iv h + 1 v h H nτ g. Here H = H V )/V is the mean of H = tr M n g h, an v h at s = 0 is the unique solution of 4) + 1 ) v h = k l h lk satisfying v h V = 0. τ M n

4 4 DAN KNOPF AND NATAŠA ŠEŠUM In components, we write 3) as 5) Nh) ) ij = 1 h) ij + R l kijg km h ml 1 gkl i l h kj + j l h ki ) + 1 i j v h H nτ g ij, where our inex convention is Rijk l = iγ l jk jγ l ik + Γl im Γm jk Γl jm Γm ik. For later use, we introuce the variant Ñ = 1 h + Rmh, ) + iv iv h + 1 v h = N + H ) nτ g. It is a classical fact [Bes87] that at any compact Einstein manifol other than the stanar sphere, the space C S T M)) of smooth sections of the bunle of symmetric covariant -tensors amits an orthogonal ecomposition C S T M)) = imiv ) C keriv) kertr) ), where C is the space of infinitesimal conformal transformations, C M n ) g. This ecomposition is also orthogonal with respect to the secon variation of ν see Theorem 1.1 of [CH15]). It is well known see Example.3 of [CHI04] or Theorem 1.4 of [CH15]) that, g FS ) is neutrally variationally stable. The neutral irection is attaine by h = ϕg C, where ϕ belongs to the first nontrivial eigenspace of the Laplacian, 6) FS + 1 τ ) ϕ = 0, ϕ V FS = 0, where the subscripts inicate that the Laplacian an volume form are those of the Fubini-Stuy metric g FS. Note that by 1), its Einstein constant is 1/τ). 3. Variational formulas In this section, we calculate the thir variation of Perelman s shrinker entropy at the Fubini Stuy metric. We begin by recalling some classical first-variation formulas [Bes87]. Proposition. Let gs) be a smooth one-parameter variation of g = g0) such that gs) = h, an set H = tr g h. Then one has: s 7) g ij = h ij, s Γ k ij = 1 i h s k j + j h k i k ) 8) h ij, Rijk l = 1 { i k h l j i l h jk j k h l i 9) s + j l h ik + Rijm l hm k Rm ijk hl m 10) R = H + iviv h) Rc, h, s 11) V = 1 s H V. },

5 DYNAMIC INSTABILITY OF UNDER RICCI FLOW 5 Assumptions an Notation. For simplicity, we aopt the following conventions., g FS ) enotes complex projective space with its canonical Fubini Stuy metric. Then one has Rc[g FS ] = τ) 1 g FS. We enote its real imension by n = im R = im C = N. By compactness, we may take gs) to be the smooth family 1) gs) = g FS + sh, s ε, ε), where h = ϕ g FS, with ϕ the unique solution of 6). Note that in the variation 1) that we consier, ϕ is the fixe function efine in 6), an hence is inepenent of s. Below, to avoi notational prolixity, we write g for gs) at arbitrary s ε, ε). Formulas shoul be assume to hol at any s unless explicitly state otherwise or ecorate with, in which case everything in sight is to be evaluate at, g0) = g FS ). If As) is any smooth one-parameter family of tensor fiels epening on s ε, ε), we write s A for the erivative evaluate at arbitrary s, an A = A s for the erivative evaluate at s = 0. We aopt similar notation for higher-orer erivatives. Thus, as note above, our selecte variation has the property that ϕ = 0, i.e., 0 = h = h =, a fact that we use frequently below. Finally, having fixe h, we simply write N = Nh) an v = v h where no confusion will result. Remark 1. We note that to prove ynamic instability of, it suffices to exhibit one unstable variation. Our choice with ϕ = 0 matches that use by Cao Zhu [CZ1], whose results we use below. Kröncke makes a ifferent choice in the proof of Proposition 9.1 of [Kro13], taking gt) = 1 + t vt) ) g FS, where vt) = ϕ/1 + tϕ). Other choices iffering to secon orer or above) are certainly possible. Remark. Our simple choice of variation, with ϕ = 0, means that we o not have the freeom to force equation 6) to hol for all s in an open set. Consequently, we o not ifferentiate that equation in what follows. We now establish various ientities neee to compute 3 s 3 ν [ gs) ]. Even though many of these ientities are well known to experts, we erive them here to keep this note as transparent an self-containe as possible.

6 6 DAN KNOPF AND NATAŠA ŠEŠUM Lemma 1. If g an h = ϕ g FS are as above, then using δ to enote the Kronecker elta function, one has: 13) g ij = ϕ g ij, s Γ k ij = 1 i ϕ δj s k + j ϕ δi k k ) 14) ϕ g ij, Rijk l = 1 i k ϕ δj s l j k ϕ δi l + j l ϕ g ik i l ) 15) ϕ g jk, 16) R = n 1) ϕ n s τ ϕ, 17) V = n s ϕ V. Proof. Formulas 13) 17) follow from 7) 11) by irect substitution. Lemma. If g an h = ϕ g FS are as above, then one has: 18) τ = 0, s 19) V = 0, s nn ) 0) H = ϕ, s V where ϕ = ϕ V. Proof. To establish 18), we recall that by Lemma.4 of [CZ1], using the fact that our metric is Einstein with Rc = τ) 1 g F S an hence that we can take f = 0 in Lemma.4 of [CZ1]), we have τ Rc, h V M = τ n R V M n Furthermore, at s = 0, equation 6) implies that Rc, h V = 1 τ H V = n τ ϕ V = 0. Equation 18) follows. Equation 19) follows from 17) an 6), because at s = 0, V = V ) = n ϕ V = 0. Finally, equation 0) follows from 19) an the computation g s CP ij h ij V = g ij ) h ij V + g ij h ij V ) N CP N n = ϕh + H ϕ) V ) = n + n ϕ V. We next compute erivatives of the Laplacian..

7 DYNAMIC INSTABILITY OF UNDER RICCI FLOW 7 Lemma 3. If g an h = ϕ g FS are as above, then at s = 0) we have 1) = ϕ + n ϕ. Proof. For any smooth possibly s-epenent) function u, using formulas 13) an 14) an collecting terms yiels { g ij i j u Γ k s ij k u) } = u ϕ u g ij Γ k ij) k u = u ϕ u + n ϕ, u, whence the result follows. We note that for u = ϕ, we have ϕ = 0.) Lemma 4. If g an h = ϕg F S are as above, then at s = 0) we have = ϕ. Proof. If u is a smooth function that is inepenent of s, then u) = g ij ) i j u + g ij ) i j u) + g ij i j u). Using the Neumann series for g FS +sϕg FS ) 1, one sees easily that g ij ) = ϕ g ij. Equation 13) gives g 1 ) = ϕg 1, an as in the proof of Lemma 3, one has Using the fact that u = 0, one has g ij i j u) = n ϕ, u. i j u) = Γ k ij) k u, where by part 1) of Lemma 8 of [AK07], we have Putting these together, we obtain Γ k ij) = h k l i h l j + j h l i l h ij ) = ϕ i ϕ δ k j + j ϕ δ k i k ϕ g ij ). u) = ϕ u ϕ n ϕ, u + ϕ n) ϕ, u = ϕ u, which finishes the proof. Lemma 5. If g an h = ϕ g FS are as above, then at s = 0, v = ϕ, an Ñ vanishes pointwise. Proof. To prove the first claim, we note that k l h lk = ϕ = ϕ/τ by 6), an hence by 4) we have + 1 ) v = ϕ τ τ CP, with vv = 0. N On the other han, by 6), we have + 1 ) ϕ = ϕ τ τ, with ϕv = 0. By uniqueness, we have v = ϕ.

8 8 DAN KNOPF AND NATAŠA ŠEŠUM To prove the secon claim, one uses the ientity R ij = τ) 1 g ij, without ifferentiating, along with 6) an the first claim to obtain Ñ ij = 1 ϕ g ij + ϕr ij i j ϕ + 1 i j v = 1 ϕ + 1 ) 1 ) τ ϕ g ij + i j v ϕ = 0. We now compute the thir variation explicitly. Lemma 6. If g an h = ϕ g FS are as above, then 3 s 3 ν [ gs) ] τ 1 ) = h, Rc + f) 4πτ) n/ τ g V, M n where the three secon erivatives on the right-han sie are given pointwise by 3), 5), an 6), respectively. Proof. By Lemma. of [CZ1], at arbitrary s ε, ε), one has s νs) = { τ4πτ) n/} h, Rc + f 1 τ g e f V = { τ4πτ) n/} g ip g jq h pq ) R ij + i j f 1 τ g ij ) e f V ), which we write schematically as A B C D. A simple calculus exercise shows that ν s) = A CP B C D + A B C D + A B C D + A B C D N CP N + A B C D + A B C D + A B C D CP N + A B C D + A B C D + A B C D. Because, g FS ) is Einstein, we have C = 0. By 18), A = 0. Thus the formula above reuces to ν = A B C D + A B C D + A B C D. Using Lemma.3 of [CZ1], an 18), we obtain C = Rc + f) 1 τ h = 1 h Rmh, ) iv iv h v = Ñ. By Lemma 5, we have C = 0, an hence ) ν s) = A B C D.

9 DYNAMIC INSTABILITY OF UNDER RICCI FLOW 9 There are three terms in C. To compute the first, we apply part 3) of Lemma 8 of [AK07] to our conformal variation, obtaining 3) R ij = ϕ ϕ n ϕ ) ) g ij + n ) ϕ i j ϕ + i ϕ j ϕ, where the right-han sie is compute with respect to the metric ata of g FS. To compute the secon term, we begin by observing that i j f) = i j f Γ k ij) k f Γ k ij) k f, As note above, the function vs) efine by v := f + H τ f ν) τ in the iscussion on page 9 of [CZ1] is given by 4) at s = 0. Moreover, by 18), we have 4) v = f + H. Then equation 4) an Lemma 5 imply that f = n ϕ. Now the fact that g FS is Einstein implies that f = 0, an so by 14) we get i j f) = i j f + ϕ, f g ij i ϕ j f j ϕ i f 5) = i j f + n ϕ g ij n ) i ϕ j ϕ. Similarly, using 18) an the fact that g = 0, we compute that the final term is 1 ) 6) τ g τ = τ g. The result follows, because D = e f V ) = V. To calculate ν, we will use the following ientity. Lemma 7. If g an h = ϕ g FS are as above, then + 1 ) f = nϕ ϕ nτ τ ϕ nτ 4τ. Proof. Recall that since f = fs) is the minimizer of ν = νs) for every s, it satisfies the following elliptic equation: τ f + f R) f + ν = 0. At s = 0, we have τ = 0 by 18) an ν = 0 by our choice of variation. Inee, at s = 0, we have ν = A B C D = 0, as seen in the proof of Lemma 6.) Because f = 0, it follows that at s = 0, { 7) τ f) + f = τ f ) } R Rτ. Again using f = 0, g 1 ) = ϕ g 1 see Lemma 4), an f = n )/ ) ϕ see Lemma 6), we get 8) f ) = ϕ f + 4 f, f + f = n ) ϕ.

10 10 DAN KNOPF AND NATAŠA ŠEŠUM Again using f = 0 an i j f) = i j f = n )/ ) i j ϕ along with 5), we compute that 9) f) = g ij ) i j f + g ij ) i j f) + g ij i j f) Finally, contracting 15) yiels = f n )ϕ ϕ + n ) ϕ. R jk = n j k ϕ 1 ϕ g jk. Using this with the ientities g 1 ) = ϕ g 1, R = g ij R ij an R = n/τ), along with equations 13) an 3), we compute that R = g ij ) R ij + g ij ) R ij + g ij R ij = ϕ R + ϕ { n ϕ + n ϕ} + n ϕ ϕ n ϕ ) + n ) ϕ ϕ + ϕ ) n ) = 4n 1)ϕ ϕ ϕ + n τ ϕ. Now combining this ientify with equations 7), 8), an 9), we obtain τ f + f = τ { n )ϕ ϕ n ) ϕ } which completes the proof. + τ { 4n 1)ϕ ϕ + n ) ϕ n τ ϕ} Rτ = nτϕ ϕ nϕ nτ τ, 4. Proof of the main result Theorem 8. If g an h = ϕ g FS are as above, then 3 s 3 ν [ gs) ] = n ϕ 3 V. 4πτ) n/ Proof. Using Lemma 6, equations 3), 5), an 6), an the fact that h = ϕg, we compute at s = 0, where e f = 1, that 3 s 3 ν [ gs) ] τ { = nϕ ϕ ϕ n 4πτ) n/ τ ϕ f + 4πτ) n/ τ nτ ) 4πτ) n/ τ ϕ V τ { = n 1) 4πτ) n/ ϕ ) + n ) ϕ ϕ + ϕ )} V nn ) ϕ ϕ n )ϕ ϕ ) V ϕ ϕ V + ϕ f V },

11 DYNAMIC INSTABILITY OF UNDER RICCI FLOW 11 where we use the consequence of 6) that ϕ V = 0. To calculate the secon integral in the last line above, we use equation 6) an Lemma 7 to see that ϕ f V = ϕ + 1 τ 1 ) f V τ = n ϕ ϕ V n CPN ϕ 3 nτ V τ 4τ ϕ V 1 ϕf V τ CP N = n ϕ ϕ V n ϕ 3 V + 1 ϕf V τ CP N = n ϕ ϕ V n ϕ 3 V + 1 ϕ f V, τ where in the last step, we use the fact that the Laplacian is self-ajoint. Thus by using 6) to evaluate ϕ, we conclue that ν [ gs) ] τ { = ϕ ϕ V + n } ϕ 3 V s 4πτ) n/ τ = n ϕ 3 V. 4πτ) n/ Our Main Theorem now follows from Theorem 8 an the following observation: Lemma 9. If N > 1, there is a solution ϕ of 6) such that ϕ 3 V 0. Proof. Our approach here is essentially the same as in [Kro13] an uses results from Part III-C of [BGM71]. We inclue it here merely for completeness. We enote by P k the vector space of polynomials on C N+1 such that fcz) = c k c k fz) an by H k P k the subspace of harmonic polynomials. Then the eigenfunctions of the k th eigenvalue of the Laplacian on, g FS ) are the restrictions of functions in H k, an one has a ecomposition 30) P k = H k r P k 1 for all integers k 1. It follows that the imension of the first nontrivial eigenspace of the the Laplacian on, g FS ) is N + 1) 1 = NN + ). Using the hypothesis N > 1, one can efine real-value functions f 1 z) = z 1 z + z 1 z, f z) = z z 3 + z z 3, f 3 z) = z 3 z 1 + z 3 z 1. We choose f = f 1 + f + f 3 an enote by ϕ its restriction to. We will show that this choice not unique in general) has the esire properties. By consiering isometries z j z j, it is easy to verify that f V = 0. One has S N f 3 = fj f j fk + 6f 1 f f 3. j=1 j=1 k j Consieration of the same isometries z j z j similarly shows that the integrals of all terms above vanish, except possibly the last. We expan that term, obtaining f 1 f f 3 = z 1 z z 3 + z 1 z z 3 + z z 3) + z z 1 z 3 + z 1z 3) + z 3 z 1 z + z 1z ).

12 1 DAN KNOPF AND NATAŠA ŠEŠUM Consieration of isometries z j iz j shows that the integrals of all terms here except the first vanish, whereupon we obtain f 3 V = 1 z 1 z z 3 V = 1 Vol S N 1). S N 1 S N 1 Now by 30), there are functions F k H k such that f 3 = F 3 +r F +r 4 F 1 +r 6 F 0, where the restriction of each F k to S N+1 is an eigenfunction of its Laplacian, with F S N 1 k V = 0 for k = 3,, 1. The fact that f 3 V > 0 thus implies that S N 1 F 0 > 0, an the same conclusion then hols for the corresponing ecomposition ϕ 3 = Φ 3 +r Φ +r 4 Φ 1 +r 6 Φ 0 into eigenvalues of the Laplacian on S N+1 /S 1. The result follows. References [AK07] Angenent, Sigur B.; Knopf, Dan Precise asymptotics of the Ricci flow neckpinch. Comm. Anal. Geom ), no. 4, [Bam14] Bamler, Richar H. Stability of hyperbolic manifols with cusps uner Ricci flow. Av. Math ), [BGM71] Berger, Marcel; Gauuchon, Paul; Mazet, Emon. Le spectre une variété Riemannienne. Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin New York 1971). [Bes87] Besse, Arthur L. Einstein manifols. Ergebnisse er Mathematik un ihrer Grenzgebiete, 10. Springer-Verlag, Berlin, [CHI04] Cao, Huai-Dong; Hamilton, Richar S.; Ilmanen, Tom. Gaussian ensities an stability for some Ricci solitons. arxiv:math/ v1. [CH15] Cao, Huai-Dong; He, Chenxu. Linear stability of Perelman s ν-entropy on symmetric spaces of compact type. J. Reine Angew. Math ), [CZ1] Cao, Huai-Dong; Zhu, Meng. On secon variation of Perelman s Ricci shrinker entropy. Math. Ann ), no. 3, [DWW05] Dai, Xianzhe; Wang, Xiaoong; Wei, Guofang. On the stability of Riemannian manifol with parallel spinors. Invent. Math ), no. 1, [DWW07] Dai, Xianzhe; Wang, Xiaoong; Wei, Guofang. On the variational stability of Kähler Einstein metrics. Comm. Anal. Geom ), no. 4, [FIK03] Felman, Mikhail; Ilmanen, Tom; Knopf, Dan. Rotationally symmetric shrinking an expaning graient Kähler Ricci solitons. J. Differential Geom ), no., [FIN05] Felman, Michael; Ilmanen, Tom; Ni, Lei. Entropy an reuce istance for Ricci expaners. J. Geom. Anal ), no. 1, [GIK0] Guenther, Christine; Isenberg, James; Knopf, Dan. Stability of the Ricci flow at Ricci-flat metrics. Comm. Anal. Geom ), no. 4, [Ham95] Hamilton, Richar S. The formation of singularities in the Ricci flow. Surveys in ifferential geometry, Vol. II Cambrige, MA, 1993), 7 136, Int. Press, Cambrige, MA, [Has1] Haslhofer, Robert Perelman s λ-functional an the stability of Ricci-flat metrics. Calc. Var. Partial Differential Equations 45 01), no. 3 4, [HM14] Haslhofer, Robert; Müller, Reto. Dynamical stability an instability of Ricci-flat metrics. Math. Ann ), no. 1, [IKS17] Isenberg, James; Knopf, Dan; Šešum, Nataša. Non-Kähler Ricci flow singularities that converge to Kähler Ricci solitons. arxiv: ) [Kno09] Knopf, Dan. Convergence an stability of locally R N -invariant solutions of Ricci flow. J. Geom. Anal ), no. 4, [Kro13] Kröncke, Klaus. Stability of Einstein metrics uner Ricci flow. Comm. Anal. Geom. In press. arxiv:131.4v). [LY10] Li, Haozhao; Yin, Hao. On stability of the hyperbolic space form uner the normalize Ricci flow. Int. Math. Res. Not. 010), no. 15, [Per0] Perelman, Grisha. The entropy formula for the Ricci flow an its geometric applications. arxiv:math.dg/011159).

13 DYNAMIC INSTABILITY OF UNDER RICCI FLOW 13 [Ses06] Šešum, Nataša. Linear an ynamical stability of Ricci-flat metrics. Duke Math. J ), no. 1, 1 6. [SSS08] Schnürer, Oliver C.; Schulze, Felix; Simon, Miles. Stability of Eucliean space uner Ricci flow. Comm. Anal. Geom ), no. 1, [SSS11] Schnürer, Oliver C.; Schulze, Felix; Simon, Miles. Stability of hyperbolic space uner Ricci flow. Comm. Anal. Geom ), no. 5, [TZ13] Tian, Gang; Zhu, Xiaohua. Convergence of the Kähler Ricci flow on Fano manifols. J. Reine Angew. Math ), [WW16] Williams, Michael Brafor; Wu, Haotian. Dynamical stability of algebraic Ricci solitons. J. Reine Angew. Math ), [Wu13] Wu, Haotian. Stability of complex hyperbolic space uner curvature-normalize Ricci flow. Geom. Deicata ), [Ye93] Ye, Rugang. Ricci flow, Einstein metrics an space forms. Trans. Amer. Math. Soc ), no., Dan Knopf) University of Texas at Austin aress: anknopf@math.utexas.eu URL: Nataša Šešum) Rutgers University aress: natasas@math.rutgers.eu URL: natasas/