Curvature estimates for Weingarten hypersurfaces in Riemannian manifolds
CURVATURE ESTIMATES FOR WEINGARTEN
HYPERSURFACES IN RIEMANNIAN MANIFOLDS
CLAUS GERHARDT
Abstract. We prove curvature estimates for general curvature func-
tions. As an application we show the existence of closed, strictly convex
hypersurfaces with prescribed curvature F , where the defining cone of
F is Γ+. F is only assumed to be monotone, symmetric, homogeneous
of degree 1, concave and of class Cm,α, m ≥ 4.
Contents
1. Introduction 1
2. Curvature estimates 5
3. Proof of Theorem 1.5 9
References 9
1. Introduction
Let N = Nn+1 be a Riemannian manifold, Ω ⊂ N open, connected and
precompact, and M ⊂ Ω a closed connected hypersurface with second fun-
damental form hij , induced metric gij and principal curvatures κi. M is said
to be a Weingarten hypersurface, if, for a given curvature function F , its
principal curvatures lie in the convex cone Γ ⊂ Rn in which the curvature
function is defined, M is then said to be admissible, and satisfies the equation
(1.1) F |M = f
where the right-hand side f is a prescribed positive function defined in Ω̄.
When proving a priori estimates for solutions of (1.1) the concavity of F
plays a central role. As usual we consider F to be defined in a cone Γ as well
as on the space of admissible tensors such that
(1.2) F (hij) = F (κi).
Date: September 2, 2021.
2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05.
Key words and phrases. curvature estimates, Weingarten hypersurface, curvature flows.
This work has been supported by the Deutsche Forschungsgemeinschaft.
http://arxiv.org/abs/0704.1021v2
2 CLAUS GERHARDT
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(SΓ ), where SΓ ⊂ T
0,2(M)
is the open set of admissible symmetric tensors with respect to the given
metric gij . The result is due to Ball, [1], see also [7, Theorem 2.1.8].
The second derivatives of F then satisfy
(1.3) F
ij,klηijηkl =
∂κi∂κj
ηiiηjj +
Fi − Fj
κi − κj
(ηij)
2 ≤ 0 ∀ η ∈ S,
where S ⊂ T 0,2(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 prove a priori estimates for the κi resulting in the fact that only
for special curvature functions for which a stronger estimate was known such
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
(1.4)
Fi − Fj
κi − κj
(ηij)
was all that is needed to obtain the stronger concavity estimates under certain
circumstances. Indeed, if the κi are labelled
(1.5) κ1 ≤ · · · ≤ κn,
then there holds:
1.1. Lemma. Let F be concave and monotone, and assume κ1 < κn, then
(1.6)
Fi − Fj
κi − κj
(ηij)
κn − κ1
(Fn − Fi)(ηni)
for any symmetric tensor (ηij), where we used coordinates such that gij = δij .
Proof. Without loss of generality we may assume that the κi satisfy the strict
inequalities
(1.7) κ1 < · · · < κn,
since these points are dense. The concavity of F implies
(1.8) F1 ≥ · · · ≥ Fn,
cf. [2, Lemma 2], where
(1.9) Fi =
the last inequality is the definition of monotonicity. The inequality then
follows immediately. �
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 κn, at
least in Riemannian manifolds, as we shall prove. Unfortunately, this doesn’t
work in Lorentzian spaces, because of a sign difference in the Gauß equations.
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 ∈ C
m,α(Γ ) ∩ C0(Γ̄ ), m ≥ 4, be symmetric, monotone,
homogeneous of degree 1, and concave such that
(1.10) F > 0 in Γ
(1.11) F |∂Γ = 0.
These conditions on the curvature function will suffice. They could have
been modified, even relaxed, e.g., by only requiring that logF is concave, but
then the condition
(1.12) F ijgij ≥ c0 > 0,
which automatically holds, if F is concave and homogeneous of degree 1,
would have been added, destroying the aesthetic simplicity of Assumption 1.2.
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(Ω̄). We
consider the curvature flow
(1.13)
ẋ = −(Φ− f̃)ν
x(0) = x0,
where Φ is Φ(r) = r and f̃ = f , x0 is the embedding of an initial admissible
hypersurface M0 of class C
m+2,α such that
(1.14) Φ− f̃ ≥ 0 at t = 0,
where of course Φ = Φ(F ) = F . We introduce the technical function Φ in the
present case only to make a comparison with former results, which all use
the notation for the more general flows, easier.
We assume that Ω̄ is covered by a Gaussian coordinate system (xα), 0 ≤
1 ≤ n, such that the metric can be expressed as
(1.15) ds̄2 = e2ψ{(dx0)2 + σijdx
and Ω̄ is covered by the image of the cylinder
(1.16) I × S0
where S0 is a compact Riemannian manifold and I = x
0(Ω̄), x0 is a global
coordinate defined in Ω̄ and (xi) are local coordinates of S0.
4 CLAUS GERHARDT
Furthermore we assume that M0 and the other flow hypersurfaces can
be written as graphs over S0. The flow should exist in a maximal time
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(T 0,1(Ω̄)) and of a constant θ > 0 such that
(1.17) 〈η, ν〉 ≥ 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 accordingly by controlled metrics from below and above, and because
the existence of such a vector field is essential for the curvature estimate.
If the flow hypersurfaces are graphs in a Gaussian coordinate system, then
such a vector field is given by
(1.18) η = (ηα) = e
ψ(1, 0, . . . , 0)
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
(1.19) 〈x, ν〉,
cf. [3, Lemma 3.5].
Then we shall prove:
1.4. Theorem. Under the assumptions stated above the principal curva-
tures κi of the flow hypersurfaces are uniformly bounded from above
(1.20) κi ≤ c,
provided there exists a strictly convex function χ ∈ C2(Ω̄). The constant c
only depends on |f |2,Ω, θ, F (1, . . . , 1), the initial data, and the estimates for
χ and those of the ambient Riemann curvature tensor in Ω̄.
Moreover, the κi will stay in a compact set of Γ .
As an application of this estimate our former results on the existence of
a strictly convex hypersurface M 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 = M1
∪ M2,
where the Mi are closed, connected strictly convex hypersurfaces of class
Cm+2,α, m ≥ 4, which can be written as graphs in a normal Gaussian coordi-
nate system covering Ω̄, and where we assume that the normal of M1 points
outside of Ω and that of M2 inside. Let 0 < f ∈ C
m,α(Ω̄), and assume that
M1 is a lower barrier for the pair (F, f) and M2 an upper barrier, then the
problem (1.1) has a strictly convex solution M ∈ Cm+2,α provided there exists
a strictly convex function χ ∈ C2(Ω̄). 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
i satisfies the evolution equation, cf. [7, Lemma 2.4.1]:
2.1. Lemma. The mixed tensor h
i satisfies the parabolic equation
(2.1)
i − Φ̇F
i;kl =
Φ̇F klhrkh
i − Φ̇Fhrih
rj + (Φ− f̃)hki h
− f̃αβx
gkj + f̃αν
i + Φ̇F
kl,rshkl;ih
+ Φ̈FiF
j + 2Φ̇F klR̄αβγδx
− Φ̇F klR̄αβγδx
rj − Φ̇F klR̄αβγδx
+ Φ̇F klR̄αβγδν
νγxδl h
i − Φ̇F R̄αβγδν
γxδmg
+ (Φ− f̃)R̄αβγδν
γxδmg
+ Φ̇F klR̄αβγδ;ǫ{ν
mj + ναx
Let η be the vector field (1.18), or any vector field satisfying (1.17), and
(2.2) ṽ = 〈η, ν〉,
then we have:
2.2. Lemma (Evolution of ṽ). The quantity ṽ satisfies the evolution equa-
(2.3)
˙̃v − Φ̇F ij ṽij =Φ̇F
j ṽ − [(Φ− f̃)− Φ̇F ]ηαβν
− 2Φ̇F ijhkjx
ηαβ − Φ̇F
ijηαβγx
− Φ̇F ijR̄αβγδν
xδjηǫx
− f̃βx
k ηαg
6 CLAUS GERHARDT
The derivation is elementary, see the proof of the corresponding lemma in
the Lorentzian case [7, Lemma 2.4.4].
Notice that ṽ is supposed to satisfy (1.17), hence
(2.4) ϕ = − log(ṽ − θ)
is well defined and there holds
(2.5) ϕ̇− Φ̇F ijϕij = −{ ˙̃v − Φ̇F
ij ṽij}
ṽ − θ
− Φ̇F ijϕiϕj .
Finally, let χ be the strictly convex function. Its evolution equation is
(2.6)
χ̇− Φ̇F ijχij = −[(Φ− f̃)− Φ̇F ]χαν
α − Φ̇F ijχαβx
≤ −[(Φ− f̃)− Φ̇F ]χαν
α − c0Φ̇F
ijgij
where c0 > 0 is independent of t.
We can now prove Theorem 1.4:
Proof of Theorem 1.4. Let ζ and w be respectively defined by
ζ = sup{ hijη
iηj : ‖η‖ = 1 },(2.7)
w = log ζ + ϕ+ λχ,(2.8)
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 x0 = x0(t0), with 0 < t0 ≤ T , be a point in M(t0)
such that
(2.9) sup
w < sup{ sup
w : 0 < t ≤ T } = w(x0).
We then introduce a Riemannian normal coordinate system (ξi) at x0 ∈
M(t0) such that at x0 = x(t0, ξ0) we have
(2.10) gij = δij and ζ = h
Let η̃ = (η̃i) be the contravariant vector field defined by
(2.11) η̃ = (0, . . . , 0, 1),
and set
(2.12) ζ̃ =
hij η̃
gij η̃
ζ̃ is well defined in neighbourhood of (t0, ξ0).
Now, define w̃ by replacing ζ by ζ̃ in (2.8); then, w̃ assumes its maximum
at (t0, ξ0). Moreover, at (t0, ξ0) we have
(2.13)
ζ = ḣnn,
and the spatial derivatives do also coincide; in short, at (t0, ξ0) ζ̃ satisfies the
same differential equation (2.1) as hnn. For the sake of greater clarity, let us
therefore treat hnn like a scalar and pretend that w is defined by
(2.14) w = log hnn + ϕ+ λχ.
CURVATURE ESTIMATES 7
From the equations (2.1), (2.5), (2.6) and (1.6), we infer, by observing the
special form of Φ, i.e., Φ(F ) = F , Φ̇ = 1, f̃ = f and using the monotonicity
and homgeneity of F
(2.15) F = F (κi) = F (
, . . . , 1)κn ≤ F (1, . . . , 1)κn
that in (t0, ξ0)
(2.16)
0 ≤ − 1
Φ̇F ijhkih
ṽ − θ
− fhnn + c(θ)Φ̇F
ijgij + λc
− λc0Φ̇F
gij − Φ̇F
ϕiϕj + Φ̇F
ij(log hnn)i(log h
κn − κ1
(Fn − Fi)(h
ni; )
2(hnn)
Similarly as in [6, p. 197], we distinguish two cases
Case 1. Suppose that
(2.17) |κ1| ≥ ǫ1κn,
where ǫ1 > 0 is small, notice that the principal curvatures are labelled ac-
cording to (1.5). Then, we infer from [6, Lemma 8.3]
(2.18) F ijhkih
F ijgijǫ
(2.19) F ijgij ≥ F (1, . . . , 1),
for a proof see e.e., [7, Lemma 2.2.19].
Since Dw = 0,
(2.20) D log hnn = −Dϕ− λDχ,
we obtain
(2.21) Φ̇F ij(log hnn)i(log h
n)j = Φ̇F
ϕiϕj + 2λΦ̇F
ϕiχj + λ
χiχj ,
where
(2.22) |ϕi| ≤ c|κi|+ c,
as one easily checks.
Hence, we conclude that κn is a priori bounded in this case.
Case 2. Suppose that
(2.23) κ1 ≥ −ǫ1κn,
8 CLAUS GERHARDT
then, the last term in inequality (2.16) is estimated from above by
(2.24)
1 + ǫ1
(Fn − Fi)(h
ni; )
2(hnn)
1 + 2ǫ1
(Fn − Fi)(h
nn; )
2(hnn)
+ c(ǫ1)Φ̇
(Fi − Fn)κ
where we used the Codazzi equation. The last sum can be easily balanced.
The terms in (2.16) containing the derivative of hnn can therefore be esti-
mated from above by
(2.25)
1− 2ǫ1
1 + 2ǫ1
nn; )
2(hnn)
1 + 2ǫ1
(h inn; )
2(hnn)
≤ Φ̇Fn
(h inn; )
2(hnn)
= Φ̇Fn‖Dϕ+ λDχ‖
= Φ̇Fn{‖Dϕ‖
2 + λ2‖Dχ‖2 + 2λ〈Dϕ,Dχ〉}.
Hence we finally deduce
(2.26)
0 ≤ −Φ̇1
ṽ − θ
+ cλ2Φ̇Fn(1 + κn)− fκn + λc
+ (c(θ) − λc0)Φ̇F
ijgij
Thus, we obtain an a priori estimate
(2.27) κn ≤ const,
if λ is chosen large enough. Notice that ǫ1 is only subject to the requirement
0 < ǫ1 <
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 M0 = M2.
The flow will exist in a maximal time interval [0, T ∗) and will stay in Ω̄.
We shall also assume that M2 is not already a solution of the problem for
otherwise the flow will be stationary from the beginning.
Furthermore, the flow hypersurfaces can be written as graphs
(3.1) M(t) = graphu(t, ·)
over S0, since the initial hypersurface has this property and all flow hypersur-
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(S0)
for some 0 < β ≤ α to a limit hypersurface M , that will be a stationary
solution, cf. [8, Section 6].
References
[1] J. M. Ball, Differentiability properties of symmetric and isotropic functions, Duke
Math. J. 51 (1984), no. 3, 699–728.
[2] Klaus Ecker and Gerhard Huisken, Immersed hypersurfaces with constant Weingarten
curvature., Math. Ann. 283 (1989), no. 2, 329–332.
[3] Claus Gerhardt, Flow of nonconvex hypersurfaces into spheres, J. Diff. Geom. 32
(1990), 299–314.
[4] , Closed Weingarten hypersurfaces in Riemannian manifolds, J. Diff. Geom. 43
(1996), 612–641, pdf file.
[5] , Hypersurfaces of prescribed Weingarten curvature, Math. Z. 224 (1997),
167–194, pdf file.
[6] , Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, J. reine
angew. Math. 554 (2003), 157–199, math.DG/0207054.
[7] , Curvature Problems, Series in Geometry and Topology, vol. 39, International
Press, Somerville, MA, 2006, 323 pp.
[8] , Curvature flows in semi-Riemannian manifolds, arXiv:0704.0236, 48 pages.
[9] Weimin Sheng, John Urbas, and Xu-Jia Wang, Interior curvature bounds for a class
of curvature equations, Duke Math. J. 123 (2004), no. 2, 235–264.
Ruprecht-Karls-Universität, Institut für Angewandte Mathematik, Im Neuen-
heimer Feld 294, 69120 Heidelberg, Germany
E-mail address: gerhardt@math.uni-heidelberg.de
URL: http://www.math.uni-heidelberg.de/studinfo/gerhardt/
http://www.math.uni-heidelberg.de/studinfo/gerhardt/Gerhardt-JDG-96.pdf
http://www.math.uni-heidelberg.de/studinfo/gerhardt/MZ224,97.pdf
http://arXiv.org/pdf/math.DG/0207054
http://arXiv.org/abs/0704.0236
	1. Introduction
	2. Curvature estimates
	3. Proof of Theorem 1.5
	References
