WHEN MAXIMUM-PRINCIPLE FUNCTIONS CEASE TO EXIST MARTINFRANZEN 5 1 Abstract. Weconsidergeometricflowequationsforcontractingandexpand- 0 ingnormalvelocities,includingpowersoftheGausscurvature,K,ofthemean 2 curvature, H, and of the norm of the second fundamental form, |A|, and ask whether-afterappropriaterescaling-closedstrictlyconvexsurfacesconverge n tospheres. Toprovethis,manyauthorsusecertainfunctionsoftheprincipal a curvatures, which we call maximum-principle functions. We show when such J functionsceasetoexistandexist,whilepresentingnewlydiscoveredmaximum- 8 principlefunctions. 2 ] G D Contents . h 1. Overview 1 t 2. Notation 2 a m 3. Contracting and expanding normal velocities Fσ 3 ξ 4. Definition and motivation of rational MPF, and MPF 5 [ 4.1. Linear operator L 7 1 4.2. α-conditions 11 v 5. Necessary conditions for the existence of MPF 18 9 5.1. Euler’s Theorem on homogeneous functions 18 5 2 5.2. No MPF for normal velocities with 00 a b for all 00 for all 00, ξ a ξ b for all 0 < a,b. Hence, Fσ is a normal velocity for all ξ ∈ R. Fσ is a contracting ξ ξ normal velocity, if σ > 0, and Fσ is an expanding normal velocity, if σ < 0. ξ Calculating the quantity β(ρ) we obtain β (ρ)=ρξ−1, Fσ ξ for all ξ ∈R. Note that β (ρ) is independent of σ. Fσ ξ Example 3.4 (Gauss curvature). Let ξ =0. Then we have β (ρ)=ρ−1, and Fσ 0 Fσ = sgn(σ)·(ab)σ/2 0 = sgn(σ)·Kσ/2. Example 3.5 (Mean curvature). Let ξ =1. Then we have β (ρ)=1, and Fσ 1 Fσ = sgn(σ)·(a+b)σ 1 = sgn(σ)·Hσ. Example 3.6 (Norm of the second fundamental form). Let ξ =2. Then we have β (ρ)=ρ, and Fσ 2 Fσ = sgn(σ)·(a2+b2)σ/2 2 = sgn(σ)·|A|σ. Example 3.7 (Trace of the second fundamental form). Let ξ =σ. Then we have β (ρ)=ρσ−1, and Fσ σ Fσ = sgn(σ)·(aσ+bσ)σ/σ σ = sgn(σ)·trAσ. MAXIMUM-PRINCIPLE FUNCTIONS 5 4. Definition and motivation of rational MPF, and MPF In the context of geometric evolution equations (1.1), d X =−Fν, dt thequestionarises,ifafterappropriaterescaling,closedstrictlyconvexsurfacesM t converge to spheres. This is also referred to as convergence to a round point or convergence to a sphere at ∞. The answer is affirmative for many normal velocities F, including the Gauss curvature flow, F2 = K, B. Andrews [2], the mean curvature flow, F1 = H, G. 0 1 Huisken [9], and the inverse Gauss curvature flow, F−1 =−1, O. Schnu¨rer [13]. 0 K FormanynormalvelocitiesF proofsrelyonthefactthatacertaingeometrically meaningful quantity w is monotone, i.e. max w is non-increasing in time. Mt In[12]and[13],O.Schnu¨rerproposescriteriaforselectingsuchmonotonequan- tities for contracting flows and for expanding flows, respectively. To date, to the author’s knowledge, all known quantities which fulfill these criteria can be used to show convergence to a round point or to a sphere at ∞. Here, we decided to work with these criteria as a definition. Their monotonicity is proven using the maximum-principle. This is why we name these quantities rational maximum- principle functions (RMPF). Definition 4.1 (RMPF). Let a and b be principal curvatures. Let w(a,b) ∈ C2(R2) be a symmetric homogeneous function of degree χ∈R\{0}. We call w a + rational maximum-principle function for a normal velocity F if (1) (a) w >0 for all 00 if F is contracting, (b) χ<0 if F is expanding. (3) (a) w <0 for all 00 for all 0 0, or if χ ≤ 0 (if χ ≥ 0). If χ ≤ 0 (if χ ≥ 0), we obtain that w is non- increasing on any self-similarly contracting (expanding) surface. So this does not implyconvergencetoaroundpoint. RMPFcondition(3)ensuresthatthequantity decreasesiftheprincipalcurvaturesapproacheachother. ByRMPFcondition(4), we check that we can apply the maximum principle to prove monotonicity. The first found monotone quantities are all rational functions, e.g. w = (a−b)2 for 6 MARTINFRANZEN F2 = K in [2], or w = (a−b)2(a+b) for F2 = |A|2 in [12]. This motivates RMPF 0 (ab) 2 condition (5). Definition 4.2 (MPF). Letaandbbeprincipalcurvatures. Letw(a,b)∈C2(R2) + beasymmetrichomogeneousfunctionofdegreeχ∈R\{0}. Wecallwamaximum- principle function for a normal velocity F if (1) (a) w >0 for all 00 if F is contracting, (b) χ<0 if F is expanding. (3) (a) w <0 for all 00 for all 00, d∈R be some constants. Set (a,b)=(ρ,1), where 0<ρ<1. If F is contracting, we assume for ρ→0 α=c, α =0, a or α= c +o(cid:0)ρ−1(cid:1), α =− c +o(cid:0)ρ−1(cid:1). ρ a ρ2 If F is expanding, we assume for ρ→0 α= c +o(cid:0)ρ−1(cid:1), α =− c +o(cid:0)ρ−1(cid:1), ρ a ρ2 or α=c+dρ +o(cid:0)ρ−1(cid:1), α =−2c+dρ +o(cid:0)ρ−1(cid:1). ρ2 a ρ2 F.SchulzeandO.Schnu¨rerpresentin[15]oneofthefirstnon-rationalmonotone quantities. They use (a−b)2(a+b)2σ w = (ab)2 MAXIMUM-PRINCIPLE FUNCTIONS 7 toshowconvergencetoaroundpointforFσ =Hσ,forall1<σ ≤5. Here,wisnot 1 rational, if 2σ ∈/ N. Therefore, in MPF Definition 4.2, we extend RMPF Definition 4.1tonotnecessarilyrationalfunctions. Wecallthemmaximum-principlefunctions (MPF). In Lemma 4.11, we prove that MPF condition (5) holds for any RMPF in the case of contracting flows Fσ, if ξ >0, or ξ =0 and 1<σ ≤2. For other Fσ, a ξ ξ similar Lemma is in progress. A first step in the this direction is Lemma 4.10. To date, all known RMPF fulfill MPF condition (5). Inthispaper,themainquestionis,whenMPFceasetoexist. Asitturnsout,this ismoreanalgebrathanadifferentialgeometryquestion. Correspondingly,inMPF condition(4)weformulatethreeinequalityconditions. InLemmas4.3through4.7, we show that these three inequality conditions are equivalent to RMPF condition (4). In condensed form, MPF Definition 4.2 is a more algebraic formulation of RMPF Definition 4.1, which is extended to what seems to be the proper class of non-rational functions. 4.1. Linear operator L. In Lemmas 4.3 through 4.7, we show that the three inequality conditions C (a,b), E (a,b), G (a,b) ≤ 0 in condition MPF (4) are w w w equivalent to RMPF condition (4) as follows: In Lemma 4.3, we show two helpful identities for the first derivatives with respect to the induced metric g , and for the derivatives with respect to the second ij fundamental form h . Next, we cite the evolution equations for g and h from ij ij ij O. Schnu¨rer [14]. We need both Lemmas in Lemma 4.5. Here, we compute the linear operator Lw, where w is a function of the principal curvatures a and b. Sometimes, we also denote them by λ and λ . In Lemma 4.6, 1 2 weciteanotherhelpfulidentityforthesecondderivativeswithrespecttothesecond fundamentalform. Finally, inLemma4.7wecomputetheconstanttermsC (a,b), w and the two gradient terms E (a,b) and G (a,b). This concludes the proof of our w w claim that condition (4) of RMPF Definition 4.1 and of MPF Definition 4.2 are equivalent. Lemma 4.3 (First derivatives). Let f be a normal velocity F or a function w of the principal curvatures λ and λ . Then we have 1 2 (4.1) fkl =fkglj, j ∂f (4.2) = −filhk. ∂g i kl The matrices (cid:0)fkl(cid:1) and (cid:16) ∂f (cid:17) are symmetric. ∂gkl Proof. We consider f = f(cid:16)hj(cid:0)(h ),(g )(cid:1)(cid:17). The matrices (h ) and (g ) are i kl kl ij ij symmetric. So we have h = 1(h +h ) and gij = 1(cid:0)gij +gji(cid:1). ij 2 ij ji 2 Now, we differentiate f with respect to h . kl ∂f fkl = ∂h kl ∂f ∂hj = i ∂hji hkl =fi ∂ 1(h +h )1(cid:0)gmj +gjm(cid:1) j ∂h 2 im mi 2 kl 8 MARTINFRANZEN =fj 1 ∂ (cid:0)h gmj +h gjm+h gmj +h gjm(cid:1) i 4∂h im im mi mi kl =fj 1(cid:0)δkδl gmj +δkδl gjm+δkδlgmj +δkδlgjm(cid:1) i 4 i m i m m i m i =fi 1(cid:0)2δkgjl+2δlgjk(cid:1) j 4 i i =1(cid:0)fkgjl+flgjk(cid:1). 2 j j Since fkl−flk =0 for all 1≤k, l≤n, the matrix (cid:0)fkl(cid:1) is symmetric and formula (4.1) follows. Next, we differentiate f with respect to g . kl ∂f = ∂f ∂ 1(h +h )1(cid:0)gmj +gjm(cid:1) ∂gkl ∂hji ∂gkl2 im mi 2 =fi ∂ 1(cid:0)h gmj +h gjm+h gmj +h gjm(cid:1) j ∂g 4 im im mi mi kl = −fi 1 (cid:0)h gmkgjl+h gjkgml+h gmkgjl+h gjkgml(cid:1) j 4 im im mi mi (cid:18) ∂gij (cid:19) use =−gikgjl ∂g kl = −fi 1 (cid:0)hkgjl+hlgjk(cid:1) j 2 i i (cid:16) (cid:17) Since ∂f − ∂f = 0 for all 1 ≤ k, l ≤ n the matrix ∂f is symmetric and ∂gkl ∂glk ∂gkl formula (4.2) follows. (cid:3) Lemma 4.4 (Evolution equations). Let X be a solution to a geometric flow equation (1.1). Then we have the following evolution equations: d (4.3) gij =2Fhij, dt (4.4) Lh =Fklhmh ·h −(cid:0)F −Fklh (cid:1)hmh +Fkl,rsh h , ij k lm ij kl i jm kl;i rs;j where L is the linear operator of RMPF Definition 4.1. Proof. We refer to O. Schnu¨rer [14]. (cid:3) Lemma 4.5 (Linear operator). Let w = w(cid:0)hj(cid:1) be a function of the principal i curvatures. Then we have Lw =wij(cid:0)h Fklhmh +hmh (cid:0)F −Fklh (cid:1)(cid:1) ij k lm i jm kl (4.5) +(cid:0)wijFkl,rs−Fijwkl,rs(cid:1)h h , kl;i rs;j where L is the linear operator of RMPF Definition 4.1. Proof. We consider w =w(cid:16)hj(cid:0)(h ),(g )(cid:1)(cid:17). i kl kl d Lw = w−Frsw dt ;rs (cid:18) (cid:19) = ∂w d (cid:0)h gmj(cid:1)−Frs ∂w (cid:0)h gmj(cid:1) ∂hj dt im ∂hj im ;r i i ;s =wi(cid:16)h˙ gmj +h g˙mj(cid:17)−Frs(cid:0)wi(cid:0)h gmj(cid:1)(cid:1) j im im j im;r ;s MAXIMUM-PRINCIPLE FUNCTIONS 9 =2F wihmjh +wigmjh˙ −Frs(cid:0)wigmjh (cid:1) j im j im j im;r ;s (use evolution equation (4.3)) =2F wijhmh +wimh˙ −Frs(cid:0)wimh (cid:1) j im im im;r ;s (cid:0)use formula (4.1): wkl =wkglj(cid:1) j =2F wijhmh +wimh˙ −Frs∂wim (cid:0)h gnl(cid:1) h −Frswimh j im im ∂hl kn ;s im;r im;rs k =2F wijhmh +wimh˙ −Frs(cid:0)wim(cid:1)kgnlh h −Frswimh j im im l kn;s im;r im;rs (cid:0)use formula (4.1): wkl =wkglj(cid:1) j (cid:16) (cid:17) =2F wijhmh +wim h˙ −Frsh −Frswim,knh h j im im im;rs im;r kn;s (cid:16) (cid:17) =2F wijhmh +wij h˙ −Fklh −Fijwkl,rsh h i jm ij ij,kl kl;i rs;j (rename indices) =wij(cid:0)Fklhmh ·h +(cid:0)F −Fklh (cid:1)hmh +Fkl,rsh h (cid:1) k lm ij kl i jm kl;i rs;j −Fijwkl,rsh h kl;i rs;j (use evolution equation (4.4)) =wij(cid:0)h Fklhmh +hmh (cid:0)F −Fklh (cid:1)(cid:1) ij k lm i jm kl +(cid:0)wijFkl,rs−Fijwkl,rs(cid:1)h h . (cid:3) kl;i rs;j Lemma 4.6 (Secondderivatives). Let f be a normal velocity F or a function w of the principal curvatures λ and λ . Then we have 1 2 (4.6) fij,klη η =(cid:88) ∂2f η η +(cid:88) ∂∂λfi − ∂∂λfj η2 ij kl ∂λ ∂λ ii jj λ −λ ij i j i j i,j i(cid:54)=j for any symmetric matrix (η ) and λ (cid:54)= λ , or λ = λ and the last term is ij 1 2 1 2 interpreted as a limit. Proof. We refer to C. Gerhardt [7]. (cid:3) Lemma 4.7 (Linear operator at a critical point). Let w = w(hj) be a symmetric i function of the principal curvatures a and b. At a critical point of w, i.e. w = 0 ;i for all i=1,2, we choose normal coordinates, i.e. g =δ and (h )=diag(a, b). ij ij ij Then we have Lw =C (a, b)+E (a, b)h2 +G (a, b)h2 , w w 11;1 w 22;2 where L is the linear operator of RMPF Definition 4.1, and α is the quantity of MPF Definition 4.2. The constant terms are C (a, b)=w a(cid:0)(cid:0)F a2+F b2(cid:1)+(F −F a−F b)a(cid:1) w a a b a b +w b(cid:0)(cid:0)F a2+F b2(cid:1)+(F −F a−F b)b(cid:1) b a b a b =F (cid:0)w a2+w b2(cid:1) a b +F w ab(a−b) a b −F w ab(a−b), b a 10 MARTINFRANZEN and the two gradient terms are E (a,b):=w (cid:0)F +2F α+F α2(cid:1) w a aa ab bb −F (cid:0)w +2w α+w α2(cid:1) a aa ab bb w F −w F +2 b a a b α2, a−b (cid:18) G (a,b):= 1 · w (cid:0)F +2F α+F α2(cid:1) w α2 b aa ab bb −F (cid:0)w +2w α+w α2(cid:1) b aa ab bb (cid:19) w F −w F +2 b a a b . a−b We obtain G (a,b)=E (b,a) for all 0

When maximum-principle functions cease to exist

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