Gradient estimates for inverse curvature flows in hyperbolic space
aa r X i v : . [ m a t h . DG ] O c t GRADIENT ESTIMATES FOR INVERSE CURVATURE FLOWSIN HYPERBOLIC SPACE
JULIAN SCHEUER
Abstract.
We prove gradient estimates for hypersurfaces in the hyperbolicspace H n +1 , expanding by negative powers of a certain class of homogeneouscurvature functions. We obtain optimal gradient estimates for hypersurfacesevolving by certain powers p > F − and smooth convergence of theproperly rescaled hypersurfaces. In particular, the full convergence result holdsfor the inverse Gauss curvature flow of surfaces without any further pinchingcondition besides convexity of the initial hypersurface. Introduction and the main result
This short note is a direct improvement of [4], in which we considered inversecurvature flows in the hyperbolic space H n +1 , n ≥ , of the form˙ x = 1 F p ν, < p < ∞ ,x (0 , M ) = M , (1.1)where(1.2) x : [0 , ∞ ) × M ֒ → H n +1 is a family of embeddings, F is a curvature function satisfying some additionalproperties specified later and M is a suitable initial hypersurface. We showed thatunder certain conditions on F and M this flow exists for all time and convergesto a well-defined smooth function after rescaling. In case p > F and M , namely F was supposed to vanish on theboundary of(1.3) Γ + = { ( κ i ) ∈ R n : κ i > ∀ i = 1 , . . . , n } and M had to satisfy a pinching condition on the oscillation, cf. [4, Thm. 1.2,(2)]. This condition was designed to ensure that the gradient of the initial convexhypersurface M was small. Then it was possible to show that it remained small.The unsatisfactory thing was that we were not able to treat powers of the Gaussiancurvature that were larger than n − without this restriction. The aim of this note isto provide gradient estimates which allow to treat (1.1) for some powers 1 < p ≤ p , where p will depend on the curvature function. In case of the Gaussian curvature, F = nK n , we will obtain p = nn − . This includes the full convergence result incase of the inverse Gauss curvature flow for surfaces with arbitrary convex initialhypersurface, a problem which to our knowledge has not been considered in theliterature before. The class of curvature functions we are able to prove the main
Date : October 20, 2018.2010
Mathematics Subject Classification.
Key words and phrases. curvature flows, inverse curvature flows, hyperbolic space.This work is being supported by the DFG. result for is closely related to the class ( K ∗ ) defined in [2, Def. 2.2.15]. We willcome back to this issue in some more detail later.Before we state the main result, let us formulate the assumptions we have to imposeon the curvature function. We have to add one assumption compared to [4, Thm.1.2].1.1. Assumption.
Let(1.4) Γ + = { ( κ i ) ∈ R n : κ i > ∀ i = 1 , . . . , n } . Assume F ∈ C ∞ (Γ + ) ∩ C (¯Γ + ) to be a • symmetric • monotone • homogeneous of degree 1 • concavecurvature function satisfying(1.5) F | ∂ Γ + = 0 , the normalization(1.6) F (1 , . . . ,
1) = n and(1.7) ∂F∂κ i κ i ≥ ǫ F ∀ i = 1 , . . . , n for some 0 < ǫ ( F ) ≤ n . We will prove the following result.1.2.
Theorem.
Let n ≥ and (1.8) x : M ֒ → H n +1 be the smooth embedding of a closed and strictly convex hypersurface M . Let F satisfy Assumption 1.1 and (1.9) 1 < p ≤ p := 11 − ǫ . Then the following statements hold.(i) There exists a unique global solution (1.10) x : [0 , ∞ ) × M ֒ → H n +1 of the curvature flow equation ˙ x = 1 F p νx (0 , M ) = M , (1.11) where ν is the outward normal to the flow hypersurfaces M t = x ( t, M ) and F is evaluated at the principal curvatures of M t . (ii) Representing M as a graph in geodesic polar coordinates, (1.12) M = { ( u (0 , x i ) , x i (0 , ξ )) : ξ ∈ M } , RADIENT ESTIMATES FOR ICF IN HYPERBOLIC SPACE 3 where u = x describes the radial distance to a point inside the convex bodyenclosed by M , we obtain that all flow hypersurfaces M t have a similar rep-resentation by a scalar function (1.13) u : [0 , ∞ ) × S n → R . The rescaled hypersurfaces ˜ M t which are described via the scalar function (1.14) ˜ u = u − tn p converge to a well-defined hypersurface in C ∞ . (iii) In case (1.15) F = nK n , where K is the Gaussian curvature, we have (1.16) p = nn − . Remark.
Note that the convergence of the functions (1.14) can not be im-proved in general. A counterexample was recently given by Hung and Wang in caseof the inverse mean curvature flow, F = H and p = 1 , cf. [3]. Also in case p > k ˚ A k = k A k − n H has the same structure in the first order terms as in case p = 1 . Remark.
Note that the n -th root of the Gaussian curvature(1.18) F = nK n fulfills Assumption 1.1 with ǫ = n . To see that the set of curvature functions sat-isfying Assumption 1.1 contains considerably more functions the reader is referredto [2, Prop. 2.2.18]. In case n = 2 this class contains, up to normalization andsmoothness, all curvature function of class ( K ∗ ) which are homogeneous of degree1 . For a definition of the class ( K ∗ ) compare [2, Sec. 2.2]. For arbitrary n ≥ F = GK a , a > , where G shares all the properties of a function belonging to class ( K ) besidesvanishing on ∂ Γ + . Furthermore note that the value ǫ in (1.7) can not be larger than n , due to thehomogeneity, which implies(1.20) F i κ i = F. Notation
Let us refresh some notation already used in [4]. We consider hypersurfaces in H n +1 , the metric of which reads in geodesic polar coordinates(2.1) d ¯ s = dr + ϑ ( r ) σ ij dx i dx j , where(2.2) ϑ ( r ) = sinh( r ) , JULIAN SCHEUER r = x denotes the geodesic distance to some point q ∈ H n +1 and σ ij is the roundmetric of the n -sphere. In such coordinates, let the hypersurface M be given as agraph over a geodesic sphere S n around q, (2.3) M = { ( u ( x ) , x i ) : ( x i ) ∈ S n } . Then the induced metric of M is given by(2.4) g ij = u i u j + ϑ ( u ) σ ij . Let(2.5) v = 1 + ϑ − σ ij u i u j ≡ | Du | , then the outward unit normal is given by(2.6) ( ν α ) = v − (1 , − ϑ − σ ij u j ) ,α = 0 , . . . , n and i = 1 , . . . , n. For curvature functions depending on the secondfundamental form and the metric,(2.7) F = F ( h kl , g kl )we let(2.8) F kl = ∂F∂h kl . For functions f defined on a manifold M, lower indices indicate covariant differen-tiation with respect to the induced metric of M, e.g. u i or v ij . A dot over a function or a tensor always indicates time derivation, e.g.(2.9) ˙ v = ddt v, whereas a prime always denotes derivation with respect to a direct argument. Forexample, if f = f ( u ) , then(2.10) f ′ = ddu f and(2.11) ˙ f = f ′ ˙ u. Note that this notation partially deviates from those in [1] and [2].3.
Rough outline of the proof
Let us shortly explain, which ingredient of our proof is the crucial one compared tothe proof in [4]. In [4, Sec. 3] we proved the longtime existence of the solution to(1.1) by successively proving the existence of spherical barriers and of bounds on v and the second fundamental form. This part of the proof is not affected by ourrelaxed assumptions at all, compare [4, Thm. 1.2 (1)]. Thus we have a longtimegraph representation of the flow hypersurfaces,(3.1) M t = { ( u ( t, x i ) , x i ( t, ξ )) : ξ ∈ M } . In order to derive convergence of the flow, we started the decay estimates by provingthat the quantity v converges to 1 exponentially fast. In case p > v was sufficiently small initially, due to the positively signed second orderterm in [4, equ. (4.3)]. As we will show in Section 4, this restriction is not necessaryin the situations described in Theorem 1.2. It turned out to be possible to furtherexploit the first negatively signed term in [4, equ. (4.3)] by using a better suitedarrangement of the terms involved. We will be able to exploit this extra term with RADIENT ESTIMATES FOR ICF IN HYPERBOLIC SPACE 5 the help of the additional assumptions in Assumption 1.1 to derive exponentialdecay of v − . In order to obtain the optimal gradient estimate(3.2) v − ≤ ce − tnp and to conclude the final convergence result we observe that the further argumentsin [4] do not depend on the pinching restriction at all and the proofs of the furtherdecay estimates apply literally, once one has proven decay of v − . Proof of the main theorem
As mentioned in Section 3 it suffices to prove decay of v − . The following propo-sition holds.4.1.
Proposition.
Let x be the solution of (1.1) under Assumption 1.1 with strictlyconvex initial hypersurface M and (4.1) 1 < p ≤ − ǫ . Let u be a corresponding graph representation of the M t . Then the quantity v =1 + | Du | satisfies the decay estimate (4.2) v − ≤ ce − λt , with suitable positive constants c, λ which depend on n, M , p and F. Proof.
The flow (1.1) is of the form(4.3) ˙ x = − Φ( F ) ν, where(4.4) Φ( r ) = − r − p . According to [1, (5.28)] for such a flow we have˙ v − Φ ′ F ij v ij = − Φ ′ F ij h ik h kj v − v − Φ ′ F ij v i v j + 2Φ ′ F ij v i u j ¯ Hn − Φ ′ F ij g ij ¯ H n v − Φ ′ F ij u i u j ¯ H ′ n v + ¯ Hn (Φ − Φ ′ F ) | Du | + 2Φ ′ F ¯ Hn v = − Φ ′ F ij (cid:18) h ik h kj − Hn h ij + ¯ H n g ij (cid:19) v + (cid:18) − p (cid:19) Φ ′ F ¯ Hn v − ′ F ¯ Hn v + (cid:18) p (cid:19) Φ ′ F ¯ Hn − Φ ′ F ij u i u j ¯ H ′ n v − v − Φ ′ F ij v i v j + 2Φ ′ F ij v i u j ¯ Hn . (4.5)Define(4.6) w = ( v − e λt , λ > , and suppose for 0 < T < ∞ that(4.7) sup [0 ,T ] × M w = w ( t , ξ ) ≥ , t > . JULIAN SCHEUER
Due to [1, (5.29)] we have at ( t , ξ )(4.8) 0 = v i = − v h ki u k + v ¯ Hn u i . Choose Riemannian normal coordinates around ( t , ξ ), in which(4.9) g ij = δ ij , h ij = κ i δ ij . Since v ( t , ξ ) > , there exists an index j ∈ { , . . . , n } , such that u j = 0 and thus(4.10) κ j = v − ¯ Hn .
Noting that in the hyperbolic space the mean curvature ¯ H of the slices { x = u } satisfies(4.11) ¯ Hn = coth u, that(4.12) ∂F∂κ i ≥ ǫ Fκ i ∀ ≤ i ≤ n and that in our present coordinate system we have(4.13) F ii = ∂F∂κ i , cf. [2, Lemma 2.1.9], we obtain from (4.5) at the point ( t , ξ ) that0 ≤ λw − Φ ′ F jj ¯ H n ( v − v e λt + (cid:18) − p (cid:19) Φ ′ F ¯ Hn v e λt − ′ F ¯ Hn ve λt + (cid:18) p (cid:19) Φ ′ F ¯ Hn e λt − Φ ′ F ij u i u j (cid:18) − ¯ H n (cid:19) ve λt ≤ λw − ǫ Φ ′ F ¯ Hn ( v − e λt + (cid:18) − p (cid:19) Φ ′ F ¯ Hn v e λt − ′ F ¯ Hn ve λt + (cid:18) p (cid:19) Φ ′ F ¯ Hn e λt − Φ ′ F ij u i u j (cid:18) − ¯ H n (cid:19) ve λt = λw + (cid:18) − p − ǫ (cid:19) Φ ′ F ¯ Hn v e λt − (2 − ǫ ) Φ ′ F ¯ Hn ve λt + (cid:18) p − ǫ (cid:19) Φ ′ F ¯ Hn e λt + ce ( λ − np ) t ≤ λw − (1 − ǫ ) Φ ′ F ¯ Hn w + (cid:18) − p − ǫ (cid:19) Φ ′ F ¯ Hn vw + ce ( λ − np ) t < λ and large t , where we also used(4.15) F + F − + v ≤ c, (4.16) ¯ Hn − ≤ ce − tnp and that the κ i range in a compact set of Γ + . We derived those facts in [4, Cor.3.6, Lemma 3.7, Prop. 3.10 and Prop. 3.11]. This is a contradiction and thus w isbounded. (cid:3) RADIENT ESTIMATES FOR ICF IN HYPERBOLIC SPACE 7
Remark.
In case of the Gaussian curvature(4.17) F = nK n we obtain(4.18) 11 − ǫ = nn − , which proves part (iii) of Theorem 1.2. References
1. Claus Gerhardt,
Closed Weingarten hypersurfaces in space forms , Geom. Anal. Calc. Var.(1996), 71–98.2. Claus Gerhardt,
Curvature problems , Series in Geometry and Topology, vol. 39, InternationalPress of Boston Inc., 2006.3. Pei-Ken Hung and Mu Tao Wang,
Inverse mean curvature flows in the hyperbolic 3-spacerevisited , to appear in Calc. Var. Partial Differ. Equ., 2014, arxiv:1406.1768v2.4. Julian Scheuer,
Non-scale-invariant inverse curvature flows in hyperbolic space , Calc. Var.Partial Differ. Equ. (2014), doi:10.1007/s00526–014–0742–9.
Ruprecht-Karls-Universit¨at, Institut f¨ur Angewandte Mathematik, Im NeuenheimerFeld 294, 69120 Heidelberg, Germany
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