Curvature estimates for Weingarten hypersurfaces in Riemannian manifolds

0.13 MB·English

by  Claus Gerhardt

CURVATURE ESTIMATES FOR WEINGARTEN
HYPERSURFACES IN RIEMANNIAN MANIFOLDS
8
0
0
CLAUSGERHARDT
2

n
a Abstract. We prove curvature estimates for general curvature func-
J tions. Asanapplicationweshowtheexistenceofclosed,strictlyconvex
hypersurfaces withprescribedcurvature F,wherethedefiningconeof
5
2 F isΓ+. F isonlyassumedtobemonotone,symmetric,homogeneous
ofdegree1,concaveandofclassCm,α,m≥4.

]
G
D
.
h Contents
t
a 1. Introduction 1
m
2. Curvature estimates 5
[ 3. Proof of Theorem 1.5 9

References 9
2
v
1
2
0
1. Introduction
1
4. Let N = Nn+1 be a Riemannian manifold, Ω ⊂ N open, connected and
0 precompact, and M ⊂ Ω a closed connected hypersurface with second fun-
7 damentalformh , inducedmetric g andprincipalcurvaturesκ . M issaid
ij ij i
0
to be a Weingarten hypersurface, if, for a given curvature function F, its
:
v principal curvatures lie in the convex cone Γ ⊂ Rn in which the curvature
Xi functionisdefined,M isthensaidtobeadmissible, andsatisfiestheequation
r (1.1) F =f
a |M
where the right-hand side f is a prescribed positive function defined in Ω¯.
When proving a priori estimates for solutions of (1.1) the concavity of F
playsacentralrole. AsusualweconsiderF tobe definedinaconeΓ aswell
as on the space of admissible tensors such that
(1.2) F(h )=F(κ ).
ij i
Date:February5,2008.
2000 Mathematics Subject Classification. 35J60, 53C21,53C44,53C50, 58J05.
Keywordsandphrases. curvatureestimates,Weingartenhypersurface,curvatureflows.
ThisworkhasbeensupportedbytheDeutscheForschungsgemeinschaft.
1
2 CLAUSGERHARDT
Notice that curvature functions are always assumed to be symmetric and if
F ∈Cm,α(Γ), 2≤m, 0<α<1, then F ∈Cm,α(S ), where S ⊂T0,2(M)
Γ Γ
is the open set of admissible symmetric tensors with respect to the given
metric g . The result is due to Ball, [1], see also [7, Theorem 2.1.8].
ij
The second derivatives of F then satisfy
∂2F F −F
(1.3) Fij,klηijηkl =X ηiiηjj +X i j(ηij)2 ≤0∀η ∈S,
∂κ ∂κ κ −κ
i j i j
i,j i6=j
where S ⊂ T0,2(M) is the space of symmetric tensors, if F is concave in Γ,
cf. [4, Lemma 1.1].
However, a mere non-positivity of the right-hand side is in general not
sufficient to provea prioriestimates for the κ resulting in the fact that only
i
forspecialcurvaturefunctions forwhichastrongerestimate wasknownsuch
a priori estimates could be derived and the problem (1.1) solved, if further
assumptions are satisfied.
Sheng et al. then realized in [9] that the term
F −F
(1.4) X i j(ηij)2
κ −κ
i j
i6=j
wasallthatisneededtoobtainthestrongerconcavityestimatesundercertain
circumstances. Indeed, if the κ are labelled
i
(1.5) κ ≤···≤κ ,
1 n
then there holds:
1.1.Lemma. Let F be concave and monotone, and assume κ <κ , then
1 n
n
F −F 2
(1.6) X i j(ηij)2 ≤ X(Fn−Fi)(ηni)2
κ −κ κ −κ
i j n 1
i6=j i=1
foranysymmetrictensor(η ),whereweusedcoordinatessuchthatg =δ .
ij ij ij
Proof. Withoutlossofgeneralitywemayassumethattheκ satisfythestrict
i
inequalities
(1.7) κ <···<κ ,
1 n
since these points are dense. The concavity of F implies
(1.8) F ≥···≥F ,
1 n
cf. [2, Lemma 2], where
∂F
(1.9) F = >0;
i
∂κ
i
the last inequality is the definition of monotonicity. The inequality then
follows immediately. (cid:3)
CURVATURE ESTIMATES 3
The right-hand side of inequality (1.6) is exactly the quantity that is
needed to balance a bad technical term in the a priori estimate for κ , at
n
leastinRiemannianmanifolds,asweshallprove. Unfortunately,thisdoesn’t
workinLorentzianspaces,becauseofasigndifferenceintheGaußequations.
The assumptions on the curvature function are very simple.
1.2. Assumption. Let Γ ⊂Rn be an open, symmetric, convex cone con-
taining Γ and let F ∈Cm,α(Γ)∩C0(Γ¯), m≥4, be symmetric, monotone,
+
homogeneous of degree 1, and concave such that
(1.10) F >0 in Γ
and
(1.11) F =0.
|∂Γ
These conditions on the curvature function will suffice. They could have
beenmodified,evenrelaxed,e.g.,byonlyrequiringthatlogF isconcave,but
then the condition
(1.12) Fijg ≥c >0,
ij 0
which automatically holds, if F is concave and homogeneous of degree 1,
wouldhavebeenadded,destroyingtheaestheticsimplicityofAssumption1.2.
Our estimates apply equally well to solutions of an equation as well as to
solutions of curvature flows. Since curvature flows encompass equations, let
us state the main estimate for curvature flows.
Let Ω ⊂ N be precompact and connected, and 0 < f ∈ Cm,α(Ω¯). We
consider the curvature flow
x˙ =−(Φ−f˜)ν
(1.13)
x(0)=x ,
0
where Φ is Φ(r) =r and f˜=f, x is the embedding of an initial admissible
0
hypersurface M of class Cm+2,α such that
0
(1.14) Φ−f˜≥0 at t=0,
whereofcourseΦ=Φ(F)=F. We introducethetechnicalfunctionΦinthe
present case only to make a comparison with former results, which all use
the notation for the more general flows, easier.
We assume that Ω¯ is covered by a Gaussian coordinate system (xα), 0 ≤
1≤n, such that the metric can be expressed as
(1.15) ds¯2 =e2ψ{(dx0)2+σ dxidxj}
ij
and Ω¯ is covered by the image of the cylinder
(1.16) I ×S
0
where S is a compact Riemannian manifold and I = x0(Ω¯), x0 is a global
0
coordinate defined in Ω¯ and (xi) are local coordinates of S .
0
4 CLAUSGERHARDT
Furthermore we assume that M and the other flow hypersurfaces can
0
be written as graphs over S . The flow should exist in a maximal time
0
interval [0,T∗), stay in Ω, and uniform C1-estimates should already have
been established.
1.3. Remark. The assumption on the existence of the Gaussian co-
ordinate system and the fact that the hypersurfaces can be written as
graphs could be replaced by assuming the existence of a unit vector field
η ∈C2(T0,1(Ω¯)) and of a constant θ >0 such that
(1.17) hη,νi≥2θ
uniformly during the flow, since this assumption would imply uniform C1-
estimates, which are the requirement that the induced metric can be esti-
mated accordinglyby controlledmetrics from below and above,and because
the existence of such a vector field is essential for the curvature estimate.
IftheflowhypersurfacesaregraphsinaGaussiancoordinatesystem,then
such a vector field is given by
(1.18) η =(η )=eψ(1,0,...,0)
α
and the C1-estimates are tantamount to the validity of inequality (1.17).
In case N = Rn+1 and starshaped hypersurfaces one could also use the
term
(1.19) hx,νi,
cf. [3, Lemma 3.5].
Then we shall prove:
1.4. Theorem. Under the assumptions stated above the principal curva-
tures κ of the flow hypersurfaces are uniformly bounded from above
i
(1.20) κ ≤c,
i
provided there exists a strictly convex function χ ∈ C2(Ω¯). The constant c
only depends on |f| , θ, F(1,...,1), the initial data, and the estimates for
2,Ω
χ and those of the ambient Riemann curvature tensor in Ω¯.
Moreover, the κ will stay in a compact set of Γ.
i
As an application of this estimate our former results on the existence of
a strictly convex hypersurfaceM solving the equation (1.1), [4, 5], which we
proved for curvature functions F of class (K), are now valid for curvature
functions F satisfying Assumption 1.2 with Γ =Γ .
+
We are even able to solve the existence problem by using a curvature
flow which formerly only worked in case that the sectional curvature of the
ambient space was non-positive.
CURVATURE ESTIMATES 5
1.5. Theorem. Let F satisfy the assumptions above with Γ = Γ and
+
assume that the boundary of Ω has two components
.
(1.21) ∂Ω =M ∪M ,
1 2
where the M are closed, connected strictly convex hypersurfaces of class
i
Cm+2,α, m≥4, which can be writtenas graphs in anormal Gaussian coordi-
nate system covering Ω¯, and where we assume that the normal of M points
1
outside of Ω and that of M inside. Let 0<f ∈Cm,α(Ω¯), and assume that
2
M is a lower barrier for the pair (F,f) and M an upper barrier, then the
1 2
problem (1.1)hasastrictlyconvexsolutionM ∈Cm+2,α providedthereexists
a strictly convex function χ∈C2(Ω¯). The solution is the limit hypersurface
of a converging curvature flow.
2. Curvature estimates
Let M(t) be the flow hypersurfaces, then their second fundamental form
hj satisfies the evolution equation, cf. [7, Lemma 2.4.1]:
i
2.1. Lemma. The mixed tensor hj satisfies the parabolic equation
i
h˙j −Φ˙Fklhj =
i i;kl
Φ˙Fklh hrhj −Φ˙Fh hrj +(Φ−f˜)hkhj
rk l i ri i k
−f˜ xαxβgkj +f˜ ναhj +Φ˙Fkl,rsh h j
αβ i k α i kl;i rs;
+Φ¨F Fj +2Φ˙FklR¯ xαxβxγxδhmgrj
(2.1) i αβγδ m i k r l
−Φ˙FklR¯ xαxβxγxδhmgrj −Φ˙FklR¯ xαxβxγxδhmj
αβγδ m k r l i αβγδ m k i l
+Φ˙FklR¯ ναxβνγxδhj −Φ˙FR¯ ναxβνγxδ gmj
αβγδ k l i αβγδ i m
+(Φ−f˜)R¯ ναxβνγxδ gmj
αβγδ i m
+Φ˙FklR¯ {ναxβxγxδxǫ gmj +ναxβxγxδ xǫgmj}.
αβγδ;ǫ k l i m i k m l
Let η be the vector field (1.18), or any vector field satisfying (1.17), and
set
(2.2) v˜=hη,νi,
then we have:
2.2.Lemma (Evolutionofv˜). The quantity v˜satisfies the evolution equa-
tion
v˜˙ −Φ˙Fijv˜ =Φ˙Fijh hkv˜−[(Φ−f˜)−Φ˙F]η νανβ
ij ik j αβ
−2Φ˙Fijhkxαxβη −Φ˙Fijη xβxγνα
j i k αβ αβγ i j
(2.3)
−Φ˙FijR¯ ναxβxγxδη xǫgkl
αβγδ i k j ǫ l
−f˜ xβxαη gik.
β i k α
6 CLAUSGERHARDT
The derivationis elementary,see the proofofthe correspondinglemma in
the Lorentzian case [7, Lemma 2.4.4].
Notice that v˜ is supposed to satisfy (1.17), hence
(2.4) ϕ=−log(v˜−θ)
is well defined and there holds
1
(2.5) ϕ˙ −Φ˙Fijϕ =−{v˜˙ −Φ˙Fijv˜ } −Φ˙Fijϕ ϕ .
ij ij i j
v˜−θ
Finally, let χ be the strictly convex function. Its evolution equation is
χ˙ −Φ˙Fijχ =−[(Φ−f˜)−Φ˙F]χ να−Φ˙Fijχ xαxβ
ij α αβ i j
(2.6)
≤−[(Φ−f˜)−Φ˙F]χ να−c Φ˙Fijg
α 0 ij
where c >0 is independent of t.
0
We can now prove Theorem 1.4:
Proof of Theorem 1.4. Let ζ and w be respectively defined by
(2.7) ζ =sup{h ηiηj: kηk=1},
ij
(2.8) w =logζ+ϕ+λχ,
where λ > 0 is supposed to be large. We claim that w is bounded, if λ is
chosen sufficiently large.
Let 0 < T < T∗, and x = x (t ), with 0 < t ≤ T, be a point in M(t )
0 0 0 0 0
such that
(2.9) supw <sup{supw: 0<t≤T }=w(x ).
0
M0 M(t)
We then introduce a Riemannian normal coordinate system (ξi) at x ∈
0
M(t ) such that at x =x(t ,ξ ) we have
0 0 0 0
(2.10) g =δ and ζ =hn.
ij ij n
Let η˜=(η˜i) be the contravariantvector field defined by
(2.11) η˜=(0,...,0,1),
and set
h η˜iη˜j
(2.12) ζ˜= ij .
g η˜iη˜j
ij
ζ˜is well defined in neighbourhood of (t ,ξ ).
0 0
Now, define w˜ by replacing ζ by ζ˜in (2.8); then, w˜ assumes its maximum
at (t ,ξ ). Moreover,at (t ,ξ ) we have
0 0 0 0
(2.13) ζ˜˙ =h˙n,
n
andthespatialderivativesdoalsocoincide;inshort,at(t ,ξ )ζ˜satisfiesthe
0 0
same differential equation (2.1) as hn. For the sake of greater clarity, let us
n
therefore treat hn like a scalar and pretend that w is defined by
n
(2.14) w=loghn+ϕ+λχ.
n
CURVATURE ESTIMATES 7
Fromthe equations(2.1),(2.5), (2.6)and(1.6), weinfer,by observingthe
special form of Φ, i.e., Φ(F) = F, Φ˙ = 1, f˜= f and using the monotonicity
and homgeneity of F
(2.15) F =F(κ )=F(κ1,...,1)κ ≤F(1,...,1)κ
i κn n n
that in (t ,ξ )
0 0
θ
0≤−1Φ˙Fijh hk −fhn+c(θ)Φ˙Fijg +λc
2 ki kv˜−θ n ij
−λc Φ˙Fijg −Φ˙Fijϕ ϕ +Φ˙Fij(loghn) (loghn)
(2.16) 0 ij i j n i n j
n
2
+ κ −κ Φ˙X(Fn−Fi)(hni;n)2(hnn)−1.
n 1
i=1
Similarly as in [6, p. 197], we distinguish two cases
Case 1. Suppose that
(2.17) |κ |≥ǫ κ ,
1 1 n
where ǫ > 0 is small, notice that the principal curvatures are labelled ac-
1
cording to (1.5). Then, we infer from [6, Lemma 8.3]
(2.18) Fijh hk ≥ 1Fijg ǫ2κ2,
ki j n ij 1 n
and
(2.19) Fijg ≥F(1,...,1),
ij
for a proof see e.e., [7, Lemma 2.2.19].
Since Dw =0,
(2.20) Dloghn =−Dϕ−λDχ,
n
we obtain
(2.21) Φ˙Fij(loghn) (loghn) =Φ˙Fijϕ ϕ +2λΦ˙Fijϕ χ +λ2Φ˙Fijχ χ ,
n i n j i j i j i j
where
(2.22) |ϕ |≤c|κ |+c,
i i
as one easily checks.
Hence, we conclude that κ is a priori bounded in this case.
n
Case 2. Suppose that
(2.23) κ ≥−ǫ κ ,
1 1 n
8 CLAUSGERHARDT
then, the last term in inequality (2.16) is estimated from above by
n
2
1+ǫ Φ˙X(Fn−Fi)(hni;n)2(hnn)−2 ≤
1
i=1
n
2
(2.24) 1+2ǫ Φ˙X(Fn−Fi)(hnn;i)2(hnn)−2
1
i=1
n−1
+c(ǫ1)Φ˙ X(Fi−Fn)κ−n2
i=1
where we used the Codazzi equation. The last sum can be easily balanced.
The terms in (2.16) containing the derivative of hn can therefore be esti-
n
mated from above by
n
1−2ǫ
− 1+2ǫ1Φ˙XFi(hnn;i)2(hnn)−2
1
i=1
n
2
+ 1+2ǫ Φ˙FnX(hnn;i)2(hnn)−2
1
i=1
(2.25)
n
≤Φ˙FnX(hnn;i)2(hnn)−2
i=1
=Φ˙F kDϕ+λDχk2
n
=Φ˙F {kDϕk2+λ2kDχk2+2λhDϕ,Dχi}.
n
Hence we finally deduce
θ
0≤−Φ˙1F κ2 +cλ2Φ˙F (1+κ )−fκ +λc
(2.26) 2 n nv˜−θ n n n
+(c(θ)−λc )Φ˙Fijg
0 ij
Thus, we obtain an a priori estimate
(2.27) κ ≤const,
n
if λ is chosenlargeenough. Notice that ǫ is only subjectto the requirement
1
0<ǫ < 1. (cid:3)
1 2
2.3. Remark. Since the initial condition F ≥ f is preserved under the
flow, a simple application of the maximum principle, cf. [4, Lemma 5.2], we
conclude that the principal curvatures of the flow hypersurfaces stay in a
compact subset of Γ.
2.4. Remark. These a priori estimates are of course also valid, if M is a
stationary solution.
CURVATURE ESTIMATES 9
3. Proof of Theorem 1.5
We consider the curvature flow (1.13) with initial hypersurface M =M .
0 2
The flow will exist in a maximal time interval [0,T∗) and will stay in Ω¯.
We shall also assume that M is not already a solution of the problem for
2
otherwise the flow will be stationary from the beginning.
Furthermore, the flow hypersurfaces can be written as graphs
(3.1) M(t)=graphu(t,·)
overS ,sincetheinitialhypersurfacehasthispropertyandallflowhypersur-
0
faces are supposed to be convex, i.e., uniform C1-estimates are guaranteed,
cf. [4].
The curvature estimates from Theorem 1.4 ensure that the curvature op-
erator is uniformly elliptic, and in view of well-known regularity results we
then conclude that the flow exists for all time and converges in Cm+2,β(S )
0
for some 0 < β ≤ α to a limit hypersurface M, that will be a stationary
solution, cf. [8, Section 6].
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Ruprecht-Karls-Universita¨t,Institutfu¨rAngewandteMathematik,ImNeuen-
heimerFeld 294,69120Heidelberg,Germany
E-mail address: [email protected]
URL:http://www.math.uni-heidelberg.de/studinfo/gerhardt/