Rigidity of the Sphere in Radially Symmetric Spaces
Xi Sisi Shen
March 9, 2016
Abstract
In this short note, we give a proof of a rigidity theorem for star-shaped imbeddings of the sphere in a radially symmetric space. Furthermore, we outline how
this proof can be extended to yield the result for general compact hypersurfaces
in radially symmetric spaces, subject to scalar curvature conditions. We will use
the theory of elliptic partial differential equations. The method is inspired by the
approach in [?].
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Derivation of the Main Equation
The main observation is that in a given radially symmetric space, the distance function
ρ from the central point satisfies the identity:
∇i ∇j ρ =
φ0 (ρ)
(gij − ∇i ρ∇j ρ)
φ(ρ)
(1)
where φ(ρ) is the ratio of the metric on the sphere of radius ρ in X to the standard
metric on the unit sphere in Euclidean space. Restricting this equality to a hypersurface
H in X, we obtain:
H
∇H
i ∇j ρ + hij (∇ρ · nH ) =
φ0 (ρ) H
H
(g − ∇H
i ρ∇j ρ)
φ(ρ) ij
(2)
where ∇ρ · nH is the derivative to the function ρ in X in the direction normal to H, and
the covariant derivatives are taken relative to the metric structure of H. Observe that
(∇ρ · nH )2 = 1 − (∇H ρ)2 . hij is the second fundamental form of the hypersurface in X.
We now rearrange and take the second symmetric function σ2 of both sides to obtain:
H
σ2 (∇H
i ∇j ρ −
φ0 (ρ) H
H
H 2
(g − ∇H
i ρ∇j ρ)) = (1 − (∇ ρ) )σ2 (hij ) =
φ(ρ) ij
= (1 − (∇H ρ)2 )(RH − RX + 2RX (nH , nH ))
(3)
where the terms on the right denote, respectively, the scalar curvature of H, the scalar
curvature of X, and the Ricci tensor of X, applied to the normal of the hypersurface H.
The Ricci tensor can be explicitly computed from φ(ρ), but what is important to us, is
that by radial symmetry it can be expressed as Rij = α(ρ)gij + β(ρ)∇i ρ∇j ρ for some
functions α and β. Hence the term on the right can be written as
(1 − (∇H ρ)2 )(RH − nα(ρ) − β(ρ) + 2α(ρ) + 2β(ρ)(1 − (∇H ρ)2 ))
= (1 − (∇H ρ)2 )(RH − (n − 2)α(ρ) + β(ρ)(1 − 2(∇H ρ)2 )
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Now, forgetting about X and just taking φ, α, β to be some given functions of ρ, we
have an equation for a scalar potential ρ on the hypersurface H, given purely in terms
of metric invariants of H. For any abstract manifold M of dimension n − 1 we can
consider the corresponding set of solutions to this equation. Any isometric imbedding
of M into X will give a solution to this equation. However not every solution necessarily corresponds to an imbedding. Further more, if two imbeddings yield the same
solution, then they are congruent. This can be seen from the following observation. Let
ρmax be the radius of the smallest sphere that contains both imbeddings. Then, both
hypersurfaces must be tangent to this sphere at the same point of M . We can rotate
one of the imbeddings, so that these two points and the tangent spaces at these points
overlap. Now since ρ uniquely determines the second fundamental form, then, after this
rotation, the two imbeddings will coincide. Hence, any upper bound to the number of
solutions to this equation will yield an upper bound for the number of its imbeddings
into X.
For the rest of this note, we will consider equation (??), as defined on an abstract
manifold M . This equation is fully non-linear and elliptic, when the right hand side is
positive for all points on the manifold. This can be seen from the following. Denote
φ0 (ρ) H
H
H
H
lij = ∇H
i ∇j ρ− φ(ρ) (gij −∇i ρ∇j ρ). Then the derivative of the equation, with respect
ij
to the Hessian is lij − lij g = l − T r(l). Assuming that the right hand side is always
positive, then σ2 (l) is always positive. Hence the square of the trace of l is always greater
than the sum of the squares of its eigenvalues. This in particular implies that the trace
is always greater in norm than any individual eigenvalue. This implies that l − T r(l) is
definite.
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Rigidity of Star-Shaped Imbeddings of the
Sphere
Now suppose that we are given a star-shaped imbedding of the sphere in a radially
symmetric space X. Then this must give rise to a solution ρ of equation (??). Let p
be a point on the sphere where the distance function ρ achieves a global maximum and
let Lt be a one parameter family of elements of the symmetry group of the sphere that
fix p. Then applying Lt to ρ gives a one parameter family of solution to equation (??),
all of which have the same value at p and which have vanishing gradient at p. Let us
symbolically write (??) as F [ρ] = 0. Then F [Lt (ρ)] − F [ρ] = 0. Hence,
Z
0
1
d
F [θLt (ρ) + (1 − θ)ρ]dθ =
dθ
Z
0
1
d
F [θLt (ρ) + (1 − θ)ρ]dθ
dθ
(4)
By using the chain rule we will find that Lt (ρ) − ρ must solve a homogeneous linear
equation that for, sufficiently small t, will be strictly elliptic, under the boundary condition that the gradient and the value at p must vanish. It will follow from the maximum
principle (Theorem 3.3 in [?]) the only solution must be 0. Hence ρ must be invariant
under the action of Lt and more generally, it must be invariant under any symmetry of
the sphere that leaves p unchanged. Hence any solution that arises from a star-shaped
imbedding must be radially symmetric around p. By solving the appropriate ODE, we
find that there can only be only one star-shaped imbedding of the sphere in X.
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References
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