We prove gradient estimates for hypersurfaces in the hyperbolic space
$\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of
homogeneous curvature functions. We obtain optimal gradient estimates for
hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth
convergence of the properly rescaled hypersurfaces. In particular, the full
convergence result holds for the inverse Gauss curvature flow of surfaces
without any further pinching condition besides convexity of the initial
hypersurface.Comment: 7 pages. Discussions are welcom

Scheuer, Julian

Geometric Flows

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arXiv

Julian Scheuer

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GRADIENT ESTIMATES FOR INVERSE CURVATURE FLOWS IN HYPERBOLIC SPACE

Online Research @ Cardiff

© 2015 Julian Scheuer, licensee De Gruyter Open.This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.Geometric Flows 2015; 1:11–16Research Article Open AccessJulian Scheuer*Gradient estimates for inverse curvatureflows in hyperbolic spaceAbstract:Weprove gradient estimates for hypersurfaces in the hyperbolic spaceHn+1, expanding by negativepowers of a certain class of homogeneous curvature functions F. We obtain optimal gradient estimates forhypersurfaces evolving by certain powers p > 1 of F−1 and smooth convergence of the properly rescaledhypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaceswithout any further pinching condition besides convexity of the initial hypersurface.Keywords: Curvature flows, Inverse curvature flows, Hyperbolic spaceMSC: 35K55, 53C21, 53C44DOI 10.1515/geofl-2015-0002Received November 17, 2014; accepted December 10, 20141 Introduction and the main resultThis short note is a direct improvement of [4], in which we considered inverse curvature flows in the hyper-bolic spaceHn+1, n ≥ 2, of the formẋ = 1Fp ν, 0 < p < ∞,x(0,M) = M0,(1)wherex : [0,∞) ×M →˓ Hn+1 (2)is a family of embeddings, F is a curvature function satisfying some additional properties specified later andM0 is a suitable initial hypersurface. We showed that under certain conditions on F and M0 this flow existsfor all time and converges to a well-defined smooth function after rescaling. In case p > 1 we had to imposequite restrictive conditions on F and M0, namely F was supposed to vanish on the boundary ofΓ+ = {(κi) ∈ Rn : κi > 0 ∀i = 1, . . . , n} (3)andM0 had to satisfy a pinching condition on the oscillation, cf. [4, Thm. 1.2, (2)]. This conditionwasdesignedto ensure that the gradient of the initial convexhypersurfaceM0was small. Then itwaspossible to show that itremained small. The unsatisfactory thing was that wewere not able to treat powers of the Gaussian curvaturethat were larger than n−1 without this restriction. The aim of this note is to provide gradient estimates whichallow to treat (1) for some powers 1 < p ≤ p0, where p0 will depend on the curvature function. In case of theGaussian curvature, F = nK 1n , we will obtain p0 = nn−1 . This includes the full convergence result in case ofthe inverse Gauss curvature flow for surfaces with arbitrary convex initial hypersurface, a problem which toour knowledge has not been considered in the literature before. The class of curvature functions we are able*Corresponding Author: Julian Scheuer: Institut für Angewandte Mathematik, Universität Heidelberg, E-mail:scheuer@math.uni-heidelberg.deUnangemeldetHeruntergeladen am | 09.02.15 16:5912 | Julian Scheuerto prove the main result for is closely related to the class (K*) defined in [2, Def. 2.2.15]. We will come back tothis issue in some more detail later.Before we state the main result, let us formulate the assumptions we have to impose on the curvaturefunction. We have to add one assumption compared to [4, Thm. 1.2].Assumption 1.1. LetΓ+ = {(κi) ∈ Rn : κi > 0 ∀i = 1, . . . , n}. (4)Assume F ∈ C∞(Γ+) ∩ C0(Γ̄+) to be a– symmetric– monotone– homogeneous of degree 1– concavecurvature function satisfyingF|∂Γ+ = 0, (5)the normalizationF(1, . . . , 1) = n (6)and∂F∂κiκi ≥ ϵ0F ∀i = 1, . . . , n (7)for some 0 < ϵ0(F) ≤ 1n .We will prove the following result.Theorem 1.1. Let n ≥ 2 andx0 : M →˓ Hn+1 (8)be the smooth embedding of a closed and strictly convex hypersurface M0. Let F satisfy Assumption 1.1 and1 < p ≤ p0 :=11 − ϵ0. (9)Then the following statements hold.(i) There exists a unique global solutionx : [0,∞) ×M →˓ Hn+1 (10)of the curvature flow equationẋ = 1Fp νx(0,M) = M0,(11)where ν is the outward normal to the flow hypersurfaces Mt = x(t,M) and F is evaluated at the principalcurvatures of Mt .(ii) Representing M0 as a graph in geodesic polar coordinates,M0 = {(u(0, xi), xi(0, ξ )) : ξ ∈ M}, (12)where u = x0 describes the radial distance to a point inside the convex body enclosed by M0, we obtainthat all flow hypersurfaces Mt have a similar representation by a scalar functionu : [0,∞) × Sn → R. (13)The rescaled hypersurfaces M̃t which are described via the scalar functionũ = u − tnp (14)converge to a well-defined hypersurface in C∞.UnangemeldetHeruntergeladen am | 09.02.15 16:59Gradient estimates for inverse curvature flows in hyperbolic space | 13(iii) In caseF = nK1n , (15)where K is the Gaussian curvature, we havep0 =nn − 1 . (16)Remark 1.1. Note that the convergence of the functions (14) can not be improved in general. A counterexam-ple was recently given by Hung and Wang in case of the inverse mean curvature flow, F = H and p = 1, cf.[3]. Also in case p > 1 we do not expect that the convergence can be improved, since the evolution equationof the traceless second fundamental form‖Å‖2 = ‖A‖2 − 1nH2 (17)has the same structure in the first order terms as in case p = 1.Remark 1.2. Note that the n-th root of the Gaussian curvatureF = nK1n (18)fulfills Assumption 1.1 with ϵ0 = 1n . To see that the set of curvature functions satisfying Assumption 1.1 con-tains considerably more functions the reader is referred to [2, Prop. 2.2.18]. In case n = 2 this class contains,up to normalization and smoothness, all curvature functions of class (K*)which are homogeneous of degree1. For a definition of the class (K*) compare [2, Sec. 2.2]. For arbitrary n ≥ 2 it contains those 1-homogeneousfunctions which can be written asF = GKa , a > 0, (19)where G shares all the properties of a function belonging to class (K) besides vanishing on ∂Γ+.Furthermore note that the value ϵ0 in (7) can not be larger than 1n , due to the homogeneity, which impliesF iκi = F. (20)2 NotationLet us refresh somenotation already used in [4].We consider hypersurfaces inHn+1, themetric of which readsin geodesic polar coordinatesds̄2 = dr2 + ϑ2(r)σijdxidxj , (21)whereϑ(r) = sinh(r), (22)r = x0 denotes the geodesic distance to some point q ∈ Hn+1 and σij is the round metric of the n-sphere. Insuch coordinates, let the hypersurface M be given as a graph over a geodesic sphere Sn around q,M = {(u(x), xi) : (xi) ∈ Sn}. (23)Then the induced metric of M is given bygij = uiuj + ϑ2(u)σij . (24)Letv2 = 1 + ϑ−2σijuiuj ≡ 1 + |Du|2, (25)then the outward unit normal is given by(να) = v−1(1, −ϑ−2σijuj), (26)UnangemeldetHeruntergeladen am | 09.02.15 16:5914 | Julian Scheuerα = 0, . . . , n and i = 1, . . . , n. For curvature functions depending on the second fundamental form and themetric,F = F(hkl , gkl), (27)we letFkl = ∂F∂hkl. (28)For functions f defined on a manifoldM, lower indices indicate covariant differentiation with respect tothe induced metric of M, e.g. ui or vij .A dot over a function or a tensor always indicates time derivation, e.g.v̇ = ddt v, (29)whereas a prime always denotes derivation with respect to a direct argument. For example, if f = f (u), thenf ′ = ddu f (30)andḟ = f ′u̇. (31)Note that this notation partially deviates from those in [1] and [2].3 Rough outline of the proofLet us shortly explain, which ingredient of our proof is the crucial one compared to the proof in [4]. In [4,Sec. 3] we proved the longtime existence of the solution to (1) by successively proving the existence of spher-ical barriers and of bounds on v and the second fundamental form. This part of the proof is not affected byour relaxed assumptions at all, compare [4, Thm. 1.2 (1)]. Thus we have a longtime graph representation ofthe flow hypersurfaces,Mt = {(u(t, xi), xi(t, ξ )) : ξ ∈ M}. (32)In order to derive convergence of the flow, we started the decay estimates by proving that the quantityv converges to 1 exponentially fast. In case p > 1 this was only possible if v was sufficiently small initially,due to the positively signed second order term in [4, equ. (4.3)]. As we will show in section 4, this restrictionis not necessary in the situations described in Theorem 1.1. It turned out to be possible to further exploit thefirst negatively signed term in [4, equ. (4.3)] by using a better suited arrangement of the terms involved. Wewill be able to exploit this extra term with the help of the additional assumptions in Assumption 1.1 to deriveexponential decay of v − 1.In order to obtain the optimal gradient estimatev − 1 ≤ ce−2tnp (33)and to conclude the final convergence resultwe observe that the further arguments in [4] do not dependon thepinching restriction at all and the proofs of the further decay estimates apply literally, once one has provendecay of v − 1.4 Proof of the main theoremAs mentioned in section 3 it suffices to prove decay of v − 1. The following proposition holds.UnangemeldetHeruntergeladen am | 09.02.15 16:59Gradient estimates for inverse curvature flows in hyperbolic space | 15Proposition 4.1. Let x be the solution of (1) under Assumption 1.1 with strictly convex initial hypersurface M0and1 < p ≤ 11 − ϵ0. (34)Let u be a corresponding graph representation of the Mt . Then the quantity v2 = 1 + |Du|2 satisfies the decayestimatev − 1 ≤ ce−λt , (35)with suitable positive constants c, λ which depend on n, M0, p and F.Proof. The flow (1) is of the formẋ = −Φ(F)ν, (36)whereΦ(r) = −r−p . (37)According to [1, equ. (5.28)] for such a flow we havev̇ − Φ′F ijvij = −Φ′F ijhikhkj v − 2v−1Φ′F ijvivj + 2Φ′F ijviujH̄n− Φ′F ijgijH̄2n2 v − Φ′F ijuiujH̄′n v +H̄n (Φ − Φ′F)|Du|2 + 2Φ′F H̄n v2= −Φ′F ij(︂hikhkj − 2H̄n hij +H̄2n2 gij)︂v +(︂1 − 1p)︂Φ′F H̄n v2− 2Φ′F H̄n v +(︂1 + 1p)︂Φ′F H̄n − Φ′F ijuiujH̄′n v− 2v−1Φ′F ijvivj + 2Φ′F ijviujH̄n .(38)Definew = (v − 1)eλt , λ > 0, (39)and suppose for 0 < T < ∞ thatsup[0,T]×Mw = w(t0, ξ0) ≥ 1, t0 > 0. (40)Due to [1, equ. (5.29)] we have at (t0, ξ0)0 = vi = −v2hki uk + vH̄n ui . (41)Choose Riemannian normal coordinates around (t0, ξ0), in whichgij = δij , hij = κiδij . (42)Since v(t0, ξ0) > 1, there exists an index j ∈ {1, . . . , n}, such that uj ≠ 0 and thusκj = v−1H̄n . (43)Noting that in the hyperbolic space the mean curvature H̄ of the slices {x0 = u} satisfiesH̄n = coth u, (44)that∂F∂κi≥ ϵ0Fκi∀1 ≤ i ≤ n (45)and that in our present coordinate system we haveF ii = ∂F∂κi, (46)UnangemeldetHeruntergeladen am | 09.02.15 16:5916 | Julian Scheuercf. [2, Lemma 2.1.9], we obtain from (38) at the point (t0, ξ0) that0 ≤ λw − Φ′F jj H̄2n2(v − 1)2v eλt0 +(︂1 − 1p)︂Φ′F H̄n v2eλt0− 2Φ′F H̄n veλt0 +(︂1 + 1p)︂Φ′F H̄n eλt0 − Φ′F ijuiuj(︂1 − H̄2n2)︂veλt0≤ λw − ϵ0Φ′FH̄n (v − 1)2eλt0 +(︂1 − 1p)︂Φ′F H̄n v2eλt0− 2Φ′F H̄n veλt0 +(︂1 + 1p)︂Φ′F H̄n eλt0 − Φ′F ijuiuj(︂1 − H̄2n2)︂veλt0= λw +(︂1 − 1p − ϵ0)︂Φ′F H̄n v2eλt0 − (2 − 2ϵ0)Φ′FH̄n veλt0+(︂1 + 1p − ϵ0)︂Φ′F H̄n eλt0 + ce(λ−2np )t0≤ λw − (1 − ϵ0)Φ′FH̄n w +(︂1 − 1p − ϵ0)︂Φ′F H̄n vw + ce(λ− 2np )t0< 0(47)for small λ and large t0, where we also usedF + F−1 + v ≤ c, (48)H̄n − 1 ≤ ce− 2tnp (49)and that the κi range in a compact set of Γ+.We derived those facts in [4, Cor. 3.6, Lemma 3.7, Prop. 3.10 andProp. 3.11]. This is a contradiction and thus w is bounded.Remark 4.1. In case of the Gaussian curvatureF = nK1n (50)we obtain11 − ϵ0= nn − 1 , (51)which proves part (iii) of Theorem 1.1.References[1] Claus Gerhardt. Closed Weingarten hypersurfaces in space forms. Geom. Anal. Calc. Var., pages 71–98, 1996.[2] Claus Gerhardt. Curvature problems, volume 39 of Series in Geometry and Topology. International Press of Boston Inc., 2006.[3] Pei-Ken Hung and Mu Tao Wang. Inverse mean curvature flows in the hyperbolic 3-space revisited. Calc. Var. Partial Differ-ential Equations, 2014. doi: 10.1007/s00526-014-0780-3.[4] Julian Scheuer. Non-scale-invariant inverse curvature flows in hyperbolic space. Calc. Var. Partial Differential Equations,2014. doi: 10.1007/s00526-014-0742-9.UnangemeldetHeruntergeladen am | 09.02.15 16:59

Gradient estimates for inverse curvature flows in hyperbolic space

AbstractWe prove gradient estimates for hypersurfaces in the hyperbolic space H</jats:p

Gradient estimates for inverse curvature flows in hyperbolic space

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