Stability of hypersurfaces with constant $r$-th anisotropic mean curvature

0.18 MB·English

by  Yijun He

STABILITY OF HYPERSURFACES WITH CONSTANT R-TH
ANISOTROPIC MEAN CURVATURE
8
0
0
YIJUN HEAND HAIZHONGLI
2

n
a
J

3 Abstract. GivenapositivefunctionF onSn which satisfiesaconvexitycondition,we
2 definether-thanisotropicmeancurvaturefunctionHrF forhypersurfacesinRn+1 which
] is a generalization of theusual r-thmean curvaturefunction. Let X :M →Rn+1 be an
G n-dimensional closed hypersurfacewith HrF+1=constant,for somer with 0≤r≤n−1,
D which is a critical point for a variational problem. We show that X(M) is stable if and
. only if X(M) is theWulff shape.
h
t
a
m
[ §1. Introduction

1
Let F: Sn → R+ be a smooth function which satisfies the following convexity condi-
v
1 tion:
6
5
(1.1) (D2F +F1) > 0, ∀x∈ Sn,
3 x
.
1
where Sn denotes the standard unit sphere in Rn+1, D2F denotes the intrinsic Hessian
0
8 of F on Sn and 1 denotes the identity on T Sn, > 0 means that the matrix is positive
x
0
: definite. We consider the map
v
Xi φ: Sn → Rn+1,
(1.2)
ar x7→ F(x)x+(gradSnF)x,
its image W = φ(Sn) is a smooth, convex hypersurface in Rn+1 called the Wulff shape
F
of F (see [3], [7], [8], [10], [13], [17], [18]). We note when F ≡ 1, W is just the sphere Sn.
F
Now let X: M → Rn+1 be a smooth immersion of a closed, orientable hypersurface.
Let ν: M → Sn denotes its Gauss map, that is, ν is the unit inner normal vector of M.
Let A = D2F +F1, S = −d(φ◦ν) = −A ◦dν. S is called the F-Weingarten
F F F F
operator, and the eigenvalues of S are called anisotropic principal curvatures. Let σ be
F r
2000 Mathematics Subject Classification. Primary 53C42, 53A10; Secondary 49Q10.
Key words and phrases. Wulff shape, F-Weingarten operator, anisotropic principal curvature, r-th
anisotropic mean curvature.
ThefirstauthorwaspartiallysupportedbyYouthScienceFoundationofShanxiProvince,China(Grant
No. 2006021001).
The second authorwas partially supported by thegrant No. 10531090 of theNSFCand bySRFDP..
1
2 Y.J.HEANDH.LI
the elementary symmetric functions of the anisotropic principal curvatures λ ,λ ,··· ,λ :
1 2 n
(1.3) σ = λ ···λ (1 ≤ r ≤ n).
r i1 ir
i1<X···<ir
We set σ = 1. The r-th anisotropic mean curvature HF is defined by HF = σ /Cr, also
0 r r r n
see Reilly [15].
For each r, 0≤ r ≤ n−1, we set
(1.4) A = F(ν)σ dA .
r,F r X
ZM
The algebraic (n+1)-volume enclosed by M is given by
1
(1.5) V = hX,νidA .
X
n+1
ZM
We consider those hypersurfaces which are critical points of A restricted to those
r,F
hypersurfaces enclosing a fixed volume V. By a standard argument involving Lagrange
multipliers, this means we are considering critical points of the functional
(1.6) F = A +ΛV(X),
r,F;Λ r,F
where Λ is a constant. We will show the Euler-Lagrange equation of F is:
r,F;Λ
(1.7) (r+1)σ −Λ = 0.
r+1
So the critical points are just hypersurfaces with HF = const.
r+1
If F ≡ 1, then the function A is just the functional A = S dA which was
r,F r M r X
studied by Alencar, do Carmo and Rosenberg in [1], where H = S /Cr is the usual r-
r Rr n
th mean curvature. For such a variational problem, they call a critical immersion X of
the functional A (that is, a hypersurface with H = constant) stable if and only if
r r+1
the second variation of A is non-negative for all variations of X preserving the enclosed
r
(n+1)-volume V. They proved:
Theorem 1.1. ([1]) Suppose 0 ≤ r ≤ n−1. Let X: M → Rn+1 be a closed hypersurface
with H = constant. Then X is stable if and only if X(M) is a round sphere.
r+1
Analogously, we call a critical immersion X of the functional A stable if and only
r,F
if the second variation of A (or equivalently of F ) is non-negative for all variations
r,F r,F;Λ
of X preserving the enclosed (n+1)-volume V.
In [13], Palmer proved the following theorem (also see Winklmann [18]):
Theorem 1.2. ([13]) Let X: M → Rn+1 be a closed hypersurface with HF =constant.
1
Then X is stable if and only if, up to translations and homotheties, X(M) is the Wulff
shape.
In this paper, we prove the following theorem:
STABILITY OF HYPERSURFACES 3
Theorem 1.3. Suppose 0 ≤ r ≤ n−1. Let X: M → Rn+1 be a closed hypersurface with
HF =constant. Then, X is stable if and only if, up to translations and homotheties,
r+1
X(M) is the Wulff shape.
Remark 1.4. In the case F ≡ 1, Theorem 1.3 becomes Theorem 1.1. Theorem 1.3 gives
an affirmative answer to the problem proposed in [8].
§2. Preliminaries
Let X: M → Rn+1 be a smooth closed, oriented hypersurface with Gauss map
ν: M → Sn, that is, ν is the unit inner normal vector field. Let X be a variation of
t
X, and ν : M → Sn be the Gauss map of X . We define
t t
dX dX
t t ⊤
(2.1) ψ = h ,ν i, ξ = ( ) ,
t
dt dt
where⊤representsthetangentcomponentandψ,ξ aredependentoft. Thecorresponding
first variation of the unit normal vector is given by (see [10], [13], [18])
′
(2.2) ν = −gradψ+dν (ξ),
t t
the first variation of the volume element is (see [2], [4] or [9])
(2.3) ∂ dA = (divξ−nHψ)dA ,
t Xt Xt
and the first variation of the volume V is
′
(2.4) V (t)= ψdA ,
Xt
ZM
where grad, div, H represents the gradients, the divergence, the mean curvature with
respect to X respectively.
t
Let {E ,··· ,E } be a local orthogonal frame on Sn, let e = e (t) = E ◦ν , where
1 n i i i t
i = 1,··· ,n and ν is the Gauss map of X , then {e ,··· ,e } is a local orthogonal frame
t t 1 n
of X : M → Rn+1.
t
The structure equation of x: Sn → Rn+1 is:
dx= θ E
i i i
dE = θ E −θ x
(2.5)  i P j ij j i
 dθi = Pjθij ∧θj

dθ − θ ∧θ = −1 R˜ θ ∧θ = −θ ∧θ
ij Pk ik kj 2 kl ijkl k l i j


where θ +θ =0 and R˜P = δ δ −δ δ P.
ij ji ijkl ik jl il jk
4 Y.J.HEANDH.LI
The structure equation of X is (see [11], [12]):
t
dX = ω e
t i i i
 dνt = −P ijhijωjei
(2.6)  de = ω e + h ω ν
 i Pj ij j j ij j t

 dωi =Pjωij ∧ωjP
dω − ω ∧ω = −1 R θ ∧θ
 ij Pk ik kj 2 kl ijkl k l


where ωij +ωji = 0, Rijkl +RijPlk = 0, and Rijkl arPe the components of the Riemannian
curvature tensor of X (M) with respect to the induced metric dX ·dX . Here we have
t t t
omitted the variable t for some geometric quantities.
∗ ∗ ∗
From de = d(E ◦ν )= ν dE = ν θ e −ν θ ν , we get
i i t t i j t ij j t i t
ωP= ν∗θ
(2.7) ij t ij
∗
( νtθi = − jhijωj,
where ωij +ωji = 0, hij = hji. P
Let F: Sn → R+ be a smooth function, we denote the coefficients of covariant differ-
ential of F, gradSnF with respect to {Ei}i=1,···,n by Fi,Fij respectively.
∗ ∗
From (2.7), d(F(ν )) = ν dF = ν ( F θ ) = − (F ◦ν )h ω , thus
t t t i i i ij i t ij j
(2.8) grad(F(νt)) = − (PFi ◦νt)hijej =Pdνt(gradSnF).
ij
X
Through a direct calculation, we easily get
(2.9) dφ= (D2F +F1)◦dx= A θ E ,
ij i j
ij
X
where A is the coefficient of A , that is, A = F +Fδ .
ij F ij ij ij
Taking exterior differential of (2.9) and using (2.5) we get
(2.10) A = A = A ,
ijk jik ikj
where A denotes coefficient of the covariant differential of A on Sn.
ijk F
We define (A ◦ν ) by
ij t k
(2.11) d(A ◦ν )+ (A ◦ν )ω + (A ◦ν )ω = (A ◦ν ) ω .
ij t kj t ki ik t kj ij t k k
k k
X X X
By a direct calculation using (2.7) and (2.11), we have
(2.12) (A ◦ν ) = − h A ◦ν .
ij t k kl ijl t
l
X
We define L by
ij
de
i ⊤
(2.13) ( ) = − L e ,
ij j
dt
j
X
STABILITY OF HYPERSURFACES 5
where ⊤ denote the tangent component, then L = −L and we have (see [2], [4] or [9])
ij ji
′
(2.14) h = ψ + {h ξ +ψh h +h L +h L }.
ij ij ijk k ik jk ik kj jk ki
k
X
Let s = A h , then from (2.7) and (2.9), we have
ij k ik kj
(2.15) P d(φ◦ν )= ν∗dφ= − s ω e .
t t ij j i
ij
X
We define S by S = −d(φ◦ν) = −A ◦dν, then we have S (e ) = s e . We call
F F F F j i ij i
S to be F-Weingarten operator. From the positive definite of (A ) and the symmetry
F ij P
of (h ), we know the eigenvalues of (s ) are all real. we call them anisotropic principal
ij ij
curvatures, and denote them by λ ,··· ,λ .
1 n
Taking exterior differential of (2.15) and using (2.6) we get
(2.16) s = s ,
ijk ikj
where s denotes coefficient of the covariant differential of S .
ijk F
We have n invariants, the elementary symmetric function σ of the anisotropic prin-
r
cipal curvatures:
(2.17) σ = λ ···λ (1 ≤ r ≤ n).
r i1 in
i1X<···ir
For convenience, we set σ = 1 and σ = 0. The r-th anisotropic mean curvature HF is
0 n+1 r
defined by
n!
(2.18) HF = σ /Cr, Cr = .
r r n n r!(n−r)!
We have by use of (2.2) and (2.6)
d((A E ⊗E )◦ν )
ij i j t ′
= h(D(A E ⊗E )) ,ν i
(2.19) ij dt ij ij i j νt t
= − A (ψ + h ξ )e ⊗e ,
P ijk ijk k l kl l Pi j
where D is the Levi-CPivita connectionPon Sn.
On the other hand, we have
d((A E ⊗E )◦ν ) de de
ij i j t ′ i ⊤ j ⊤
(2.20) = {A e ⊗e +A ( ) ⊗e +A e ⊗( ) }.
dt ij i j ij dt j ij i dt
ij ij
X X
By use of (2.13), we get from (2.19) and (2.20)
d(A ◦ν )
ij t ′
(2.21) = A (t) = {−A ψ − A h ξ +A L +A L }.
dt ij ijk k ijk kl l ik kj jk ki
k l
X X
By (2.12), (2.14), (2.21) and the fact L = −L , through a direct calculation, we get
ij ji
the following lemma:
6 Y.J.HEANDH.LI
ds
ij ′
Lemma 2.1. = s (t) = {(A ψ ) +s ξ +ψs h +s L +s L }.
dt ij k ik k j ijk k ik kj kj ki ik kj
P
As M is a closed oriented hypersurface, one can find a point where all the principal
curvatures with respect to ν are positive. By the positiveness of A , all the anisotropic
F
principal curvatures are positive at this point. Using the results of G˚arding ([5]), we have
the following lemma:
Lemma 2.2. Let X: M → Rn+1 be a closed, oriented hypersurface. Assume HF > 0
r+1
holds on every point of M, then HF > 0 holds on every point of M for every k = 1,··· ,r.
k
Using the characteristic polynomial of S , σ is defined by
F r
n
(2.22) det(tI −S )= (−1)rσ tn−r.
F r
r=0
X
So, we have
(2.23) σ = 1 δj1···jrs ···s ,
r r! i1···ir i1j1 irjr
i1,···,iXr;j1,···,jr
where δj1···jr is the usual generalized Kronecker symbol, i.e., δj1···jr equals +1 (resp. -1) if
i1···ir i1···ir
i ···i are distinct and (j ···j ) is an even (resp. odd) permutation of (i ···i ) and in
1 r 1 r 1 r
other cases it equals zero.
We introduce two important operators P and T by
r r
(2.24) Pr = σrI −σr−1SF +···+(−1)rSFr, r = 0,1,··· ,n,
(2.25) T = P A , r = 0,1,··· ,n−1.
r r F
Obviously P = 0 and we have
n
(2.26) Pr =σrI −Pr−1SF = σrI +Tr−1dν, r = 1,··· ,n.
From the symmetry of A and dν, S A and dν ◦S are symmetric, so T = P A
F F F F r r F
and dν ◦P are also symmetric for each r.
r
Lemma 2.3. The matrix of P is given by:
r
(2.27) (P ) = 1 δj1···jris ···s
r ij r! i1···irj i1j1 irjr
i1,···,iXr;j1,···,jr
Proof. We prove Lemma 2.3 inductively. For r = 0, it is easy to check that (2.27) is true.
Assume (2.27) is true for r = k, then from (2.26),
(P ) = σ δi − (P ) s
k+1 ij k+1 j l k il lj
= (k+11)! P(δij11······ijkk++11δji − lδij11······ijll−−11iiljill++11······jikk++11δjjl)si1j1···sik+1jk+1
= 1 Pδj1···jk+1is ·P··s .
(k+1)! i1···ik+1j i1j1 ik+1jk+1
P
STABILITY OF HYPERSURFACES 7
(cid:3)
Lemma 2.4. For each r, we have
(i). (P ) = 0,
j r jij
(ii). tr(P S )= (r+1)σ ,
P r F r+1
(iii). tr(P )= (n−r)σ ,
r r
(iv). tr(P S2)= σ σ −(r+2)σ .
r F 1 r+1 r+2
Proof. We only prove (ii), the others are easily obtained from (2.23), (2.26) and (2.27).
Noting (j,j ) is symmetric in s ···s by (2.16) and (j,j ) is skew symmetric in
r i1j1 irjrj r
δj1···jrj , we have
i1···iri
(P ) = 1 δj1···jrjs ···s =0.
r jij (r−1)! i1···iri i1j1 irjrj
Xj i1,···,irX;j1,···,jr;j
(cid:3)
Remark2.5. WhenF = 1,Lemma2.4wasawell-knownresult(forexample,seeBarbosa-
Colares [2], Reilly [14], or Rosenberg [16]).
Since Pr−1SF is symmetric and Lij is anti-symmetric, we have
(2.28) (Pr−1)ji(skjLki+sikLkj)= 0.
i,j,k
X
From (2.16), (2.26) and (i) of Lemma 2.4, we get
(2.29) (σr)k = (σrδjk)j = (Pr)jkj + [(Pr−1)jlslk]j = (Pr−1)jisijk.
j j jl ij
X X X X
§3. First and second variation formulas of F
r,F;Λ
∞ ∞
Define the operator L :C (M) → C (M) as following:
r
(3.1) L f = [(T ) f ] .
r r ij j i
i,j
X
dσ
r ′
Lemma 3.1. dt = σr(t) = Lr−1ψ+ψhTr−1◦dνt,dνti+hgradσr,ξi.
Proof. Using (2.23), (2.28), (2.29), Lemma 2.1, Lemma 2.3 and (i) of Lemma 2.4, we have
σ′ = 1 δj1···jrs ···s s′
r (r−1)! i1,···,ir;j1,···,jr i1···ir i1j1 ir−1jr−1 irjr
′
= ijk(Pr−P1)jisij
= Pijk(Pr−1)ji[(Aikψk)j +ψsikhkj +sijkξk +skjLki+sikLkj]
= Pijk[(Pr−1)jiAikψk]j +ψ ijkl(Pr−1)jiAilhlkhkj + k(σr)kξk
= Pjk[(Tr−1)jkψk]j +ψ ijPk(Tr−1)jihikhkj + k(σr)Pkξk
= LPr−1ψ+ψhTr−1◦dνt,Pdνti+hgradσr,ξi. P
8 Y.J.HEANDH.LI
(cid:3)
Lemma 3.2. For each 0≤ r ≤ n, we have
(3.2) div(Pr(gradSnF)◦νt)+F(νt)tr(Pr ◦dνt)= −(r+1)σr+1,
and
⊤
(3.3) div(P X )+hX,ν itr(P ◦dν )= (n−r)σ .
r t r t r
Proof. From (2.6), (2.15) and Lemma 2.4,
⊤
div(Pr(gradSnF)◦νt) = div(Pr(φ◦νt) )
= ((P ) hφ◦ν ,e i)
ij r ji t i j
= − (P ) s +F(ν ) (P ) h
P ij r ji ij t ij r ji ij
= −tr(P S )−F(ν )tr(P ◦dν )
P r F t Pr t
= −(r+1)σ −F(ν )tr(P ◦dν ),
r+1 t r t
⊤
div(P X ) = ((P ) hX,e i)
r ij r ji i j
= (P ) δ + (P ) h hX,ν i
Pij r ji ij ij r ji ij t
= tr(P )−tr(P ◦dν )hX,ν i
P r r P t t
= (n−r)σ −tr(P ◦dν )hX,ν i.
r r t t
Thus, the conclusion follows. (cid:3)
Theorem 3.3. (First variational formula of A )
r,F
(3.4) A′ (t) = −(r+1) ψσ dA .
r,F r+1 Xt
ZM
′ ′
Proof. We have(F(νt)) = hgradSnF,νti, sobyuseofLemma3.1, Lemma3.2, (2.2), (2.3),
(2.8), (2.26) and Stokes formula, we have
A′ (t)= (F(ν )σ′ +(F(ν ))′σ )dA +F(ν )σ ∂ dA
r,F M t r t r Xt t r t Xt
= RM{F(νt)div(Tr−1gradψ)+F(νt)ψhTr−1◦dνt,dνti+F(νt)hgradσr,ξi
R+hσr(gradSnF)◦νt,−gradψ+dνt(ξ)i+F(νt)σr(−nHψ+divξ)}dAXt
= M{−hgrad(F(νt)),Tr−1gradψi+F(νt)ψhTr−1◦dνt,dνti
R+hF(νt)gradσr,ξi+ψdiv(σr(gradSnF)◦νt)+hσrgrad(F(νt)),ξi
−nHψF(ν )σ +F(ν )σ divξ}dA
t r t r Xt
= M{−hTr−1grad(F(νt)),gradψi+F(νt)ψhTr−1◦dνt,dνti
R+ψdiv(σr(gradSnF)◦νt)−nHψF(νt)σr}dAXt
= M ψ{div(σr(gradSnF)◦νt)+div(Tr−1grad(F(νt)))
+R F(νt)hTr−1◦dνt,dνti−nHF(νt)σr}dAXt
= M ψ{div[(σr +Tr−1◦dνt)(gradSnF)◦νt]
R+F(νt)tr[(Tr−1◦dνt+σrI)◦dνt]}dAXt
= M ψ{div(Pr(gradSnF)◦νt)+F(νt)tr(Pr ◦dνt)}dAXt
= −(r+1) ψσ dA .
R M r+1 Xt
R
STABILITY OF HYPERSURFACES 9
(cid:3)
Remark 3.4. When F = 1, Lemma 4.1 and Theorem 3.3 were proved by R. Reilly [14]
(also see [2], [4]).
From (1.6), (2.4) and (3.4), we get
Proposition 3.5. (the first variational formula). For all variations of X preserving V,
we have
(3.5) A′(t) = F′ (t) = − ψ{(r+1)σ −Λ}dA .
r r,F;Λ r+1 Xt
ZM
Hence we obtain the Euler-Lagrange equation for such a variation
(3.6) (r+1)σ −Λ = 0.
r+1
Theorem 3.6. (thesecond variational formula). LetX : M → Rn+1 bean n-dimensional
closed hypersurface, which satisfies (3.6), then for all variations of X preserving V, the
second variational formula of A at t =0 is given by
r,F
(3.7) A′′(0) = F′′ (0) = −(r+1) ψ{L ψ+ψhT ◦dν,dνi}dA ,
r r,F;Λ r r X
ZM
where ψ satisfies
(3.8) ψdA = 0.
X
ZM
Proof. Differentiating (3.5), we get (3.7) by use of (3.6). (cid:3)
We call X : M → Rn+1 to be a stable critical point of A for all variations of X
r,F
preserving V, if it satisfies (3.6) and A′′(0) ≥ 0 for all ψ with condition (3.8).
r
§4. Proof of Theorem 1.3
Firstly, we prove that if X(M) is, up to translations and homotheties, the Wulff
shape, then X is stable.
From dφ = (D2F +F1)◦dx, dφ is perpendicular to x. So ν = −x is the unit inner
normal vector. We have
(4.1) dφ= −A ◦dν = A h ω e .
F jk ki i j
ijk
X
On the other hand,
(4.2) dφ= ω e ,
i i
i
X
so we have
(4.3) s = A h = δ .
ij ik kj ij
k
X
10 Y.J.HEANDH.LI
From this, we easily get σ = Cr and σ = Cr+1, thus the Wulff shape satisfies (3.6)
r n r+1 n
with Λ = (r+1)Cr+1. Through a direct calculation, we easily know for Wulff shape,
n
(4.4) Ar′′(0) = −(r+1)Cnr−1 [div(AF gradψ)+ψhAF ◦dν,dνi]dAX,
ZM
and ψ satisfies
(4.5) ψdA = 0.
X
ZM
From Palmer [13] (also see Winklmann [18]), we know A′′(0) ≥ 0, that is, the Wulff shape
r
is stable.
Next, we prove that if X is stable, then up to translations and homotheties, X(M)
is the Wulff shape. We recall the following lemmas:
Lemma 4.1. ([7], [8]) For each r = 0,1,··· ,n − 1, the following integral formulas of
Minkowski type hold:
(4.6) (HFF(ν)+HF hX,νi)dA = 0, r =0,1,··· ,n−1.
r r+1 X
ZM
Lemma 4.2. ([7], [8], [13]) If λ = λ = ··· = λ = const 6= 0, then up to translations
1 2 n
and homotheties, X(M) is the Wulff shape.
From Lemma 4.1 and (3.8), we can choose ψ = αF(ν) + HF hX,νi as the test
r+1
function, where α = F(ν)HFdA / F(ν)dA . For every smooth function f: M →
M r X M X
R, and each r, we define:
R R
(4.7) I [f]= L f +fhT ◦dν,dνi,
r r r
Then, we have from (3.7)
(4.8) A′′(0) = −(r+1) ψI [ψ]dA .
r r X
ZM
Lemma 4.3. For each 0≤ r ≤ n−1, we have
(4.9) Ir[F ◦ν] = −hgradσr+1,(gradSnF)◦νi+σ1σr+1−(r+2)σr+2,
and
⊤
(4.10) I [hX,νi] = −hgradσ ,X i−(r+1)σ .
r r+1 r+1
Proof. From (2.8) and (2.26),
I [F ◦ν] = div{T grad(F(ν))}+F(ν)hT ◦dν,dνi
r r r
= div(Tr ◦dν(gradSnF)◦ν)+F(ν)hTr ◦dν,dνi
= div(Pr+1(gradSnF)◦ν)+F(ν)tr(Pr+1dν)−hgradσr+1,(gradSnF)◦νi
−σr+1{div(P0(gradSnF)◦ν)+F(ν)tr(P0dν)},