Stability of $C^\infty$ convex integrands

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by  Erica Boizan Batista

STABILITY OF C∞ CONVEX INTEGRANDS
ERICA BOIZAN BATISTA, HUHE HAN, AND TAKASHI NISHIMURA
Abstract. Inthispaper,itisshownthatthe setconsistingofstableconvexintegrands
Sn →R+ is open and dense in the set consisting of C∞ convex integrands with respect
to Whitney C∞ topology. Moreover, an application of the proof of this result is also
shown.
6
1
0 1. Introduction
2

n In the celebrated series [14, 15, 16, 17, 18, 19], J. Mather gave a complete answer to
a the problem on density of proper stable mappings in a surprizing form. For proper C∞
J
mappings of special type, it is natural to ask the similar question, namely to ask “Are
4
2 generic proper mappings of special type stable ?”. Such investigations, for instance, can
be found in [20] for generic projections of submanifolds, in [6] for generic projections of
]
T stable mappings and in [8, 9, 10, 11] for generic distance-squared mappings and their
G generalizations.
. Motivated by these researches, in this paper, it is investigated the density problem
h
t for C∞ convex integrands. The notion of convex integrand was firstly introduced in [23],
a
m which isdefined asfollows. For apositive integer n, let Sn be theunit sphere ofRn+1. The
[ set consisting of positive real numbers is denoted by R+. Then, a continuous function γ :
Sn → R is called a convex integrand if the boundary of the convex hull of inv(graph(γ))
1 +
v is exactly the same set as inv(graph(γ)), where graph(γ) is the set {(θ,γ(θ)) | θ ∈ Sn}
7 with respect to the polar plot expression for Rn+1−{0} and inv: Rn+1−{0} → Rn+1−{0}
4
3 is the inversion defined by inv(θ,r) = −θ, 1 . The notion of convex integrand is closely
r
6 related with the notion of Wulff shape,(cid:0)which(cid:1)was firstly introduced in [24] as a geometric
0
model of crystal at equilibrium. Integration of a convex integrand γ over Sn represents
.
1
the surface energy of the Wulff shape associated with γ. Hence, γ is called a convex
0
6 integrand. For more details on convex integrands, see for instance [22, 23].
1 Set
:
v C∞ (Sn,R ) = {γ ∈ C∞(Sn,R ) | γ is a convex integrand},
i conv + +
X where C∞(Sn,R ) is the set consisting of C∞ functions Sn → R . The set C∞(Sn,R ) is
+ + +
ar endowed with Whitney C∞ topology (for details on Whitney C∞ topology, see for instance
[2, 7]); and the set C∞ (Sn,R ) is a topological subspace of C∞(Sn,R ). Two C∞
conv + +
functions γ ,γ : Sn → R are said to be A-equivalent if there exist C∞ diffeomorphisms
1 2 +
h : Sn → Sn and H : R → R such that the equality γ = H ◦ γ ◦h−1 holds. A C∞
+ + 2 1
function γ ∈ C∞(Sn,R ) is said to be stable if the A-equivalence class A(γ) is an open
+
subset of the topological space C∞(Sn,R ). By definition, any function A-equivalent to
+
2010 Mathematics Subject Classification. 58K05,58K25.
Key words and phrases. Convex integrand, C∞ convex integrand, Stable convex integrand, Caustic,
Symmetry set, Wave front, Index of critical point, Morse inequalities.
This work is partially supported by JSPS and CAPES under the Japan–Brazil research cooperative
program and JSPS KAKENHI Grant Number 26610035.
1
2 E.B. BATISTA,H. HAN,AND T. NISHIMURA
a stable function is stable. Set
S∞(Sn,R ) = {γ ∈ C∞(Sn,R ) | γ is stable}.
+ +
By definition, S∞(Sn,R ) is open. The following proposition is one of corollaries of
+
Mather’s series [14, 15, 16, 17, 18, 19].
Proposition 1. (1) A C∞ function γ ∈ C∞(Sn,R ) is stable if and only if all
+
critical points of γ are non-degenerate and γ(θ ) 6= γ(θ ) holds for any two distinct
1 2
critical points θ ,θ ∈ Sn.
1 2
(2) The open subset S∞(Sn,R ) is dense in C∞(Sn,R ).
+ +
The assertion (2) of Proposition 1 asserts that any C∞ function γ : Sn → R can
+
be perturbed to a stable function γ by a sufficiently small perturbation; and for any
sufficiently small ε > 0, any continuous mapping Φ : (−ε,ε) → C∞(Sn,R ) such that
+
Φ(0) = γ and any two t ,t ∈ (−ε,εe), there exist C∞ diffeomorphisms h : Sn → Sn and
1 2
H : R → R such that the equality Φ(t ) = H ◦Φ(t )◦h−1 holds.
+ + 2 1
e
The main purpose of this paper is to show the following:
Theorem 1. The open subset S∞(Sn,R )∩C∞ (Sn,R ) is dense in C∞ (Sn,R ).
+ conv + conv +
Similarly as the assertion (2) of Proposition 1, Theorem 1 asserts that any C∞ convex
integrand γ : Sn → R can be perturbed to a stable convex integrand γ by a sufficiently
+
small perturbation; and for any sufficiently small ε > 0, any continuous mapping Φ :
(−ε,ε) → C∞ (Sn,R ) such that Φ(0) = γ and any two t ,t ∈ (−eε,ε), there exist
conv + 1 2
C∞ diffeomorphisms h : Sn → Sn and H : R → R such that the equality Φ(t ) =
+ + 2
H ◦Φ(t )◦h−1 holds. e
1
In Section 2, preliminaries for the proof of Theorem 1 are given. Theorem 1 is proved
in Section 3. In Section 4, an application of the proof of Theorem 1 is given.
2. Preliminaries
Let φ : Sn → Rn+1 be a C∞ embedding. Consider the family of functions F : Rn+1 ×
Sn → R defined by
1
F(v,z) = ||φ(z)−v||2.
2
Notice that F may be regarded as a mapping from Rn+1 to C∞(Sn,R) which maps each
v ∈ Rn to the function f (z) = F(v,z) ∈ C∞(Sn,R) = {g : Sn → R C∞}. The set of
v
values v for which f (z) has a degenerate critical point, denoted by Caust(φ), is called
v
the Caustic of φ (for details on caustics, for instance see [1, 2, 12, 13]). The set of values
v for which f (z) has a multiple critical value forms the Symmetry set of φ, denoted by
v
Sym(φ) (for details on symmetry sets, see for instance [3, 4, 5]). By the assertion (1)
of Proposition 1, these two sets Caust(φ) and Sym(φ) constitute the set of points v for
which the function f ∈ C∞(Sn,R) is not stable.
v
Proposition 2. Let φ : Sn → Rn+1 be a C∞ embedding. Then, Caust(φ) has Lebesgue
measure zero in Rn+1.
For the proof of Proposition 2, see [21], §6 “Manifolds in Euclidean space”.
Proposition 3. Let φ : Sn → Rn+1 be a C∞ embedding. Then, Sym(φ) has Lebesgue
measure zero in Rn+1.
STABILITY OF C∞ CONVEX INTEGRANDS 3
Proof. Since φ is an embedding, the complement of φ(Sn) constitutes two connected
components. Denote the bounded connected component by V . Set M = φ(Sn). For
φ
each θ ∈ Sn, consider the normal vector space N (M) to M at θ. Notice that N (M)
φ(θ) φ(θ)
is a 1-dimensional vector space. Thus, we can uniquely specify the unit vector n(θ) of
N (M) so that φ(θ)+εn(θ)belongstoV foranysufficiently small ε > 0. Foranyt ∈ R,
φ(θ) φ
let φ : Sn → Rn+1 be the C∞ mapping defined by φ (θ) = φ(θ)+tn(θ). The mapping φ
t t t
is called a wave front of φ (for details on wave fronts, see for instance [1, 2, 12, 13]). It is
clear that, by using wave fronts {φt}t∈R, the set Sym(φ) can be characterized as follows:
Sym(φ) = {φ (θ ) = φ (θ ) | θ 6= θ }.
t 1 t 2 1 2
t[∈R
By Proposition 2, the intersection Sym(φ) ∩ Caust(φ) is of Lebesgue measure zero.
Thus, in order to show Proposition 3, it is sufficient to show that Sym(φ) ∩ (Rn+1 −
Caust(φ)) is of Lebesgue measure zero. Take one point φ (θ ) = φ (θ ) of Sym(φ) ∩
t0 1 t0 2
(Rn+1 −Caust(φ)), where θ ,θ are two distinct point of Sn. Set x = φ (θ ) = φ (θ ),
1 2 0 t0 1 t0 2
and let U be a sufficiently small open neighborhood of x . Notice that, since Caust(φ)
0 0
is compact, U may be chosen so that U ∩Caust(φ) = ∅.
0 0
Let i be 1 or 2. For the i, define the mapping t ,θ : U → R×Sn as follows:
i i 0
(cid:16) (cid:17)
e
x = φ (θ (x)), t (x ) = t ,θ (x ) = θ .
ti(x) i i 0 0 i 0 i
(cid:16) (cid:17)
Notice that, since U ∩ Caust(φe) = ∅, both of the feollowing two are well-defined C∞
0
diffeomorphisms.
t ,θ : U → t ,θ (U ),
1 1 0 1 1 0
(cid:16) (cid:17) (cid:16) (cid:17)
e e
t ,θ : U → t ,θ (U ).
2 2 0 2 2 0
(cid:16) (cid:17) (cid:16) (cid:17)
Set T = t −t . Then, it is clearethat Sym(φ)∩Ue= T−1(0).
1 2 0
For any i = 1,2, let ∇t (x ) be the gradient vector of t at x . Since both t ,t are
i 0 i 0 1 2
non-singular functions, it follows that neither ∇t (x ) nor ∇t (x ) is the zero vector.
1 0 2 0
Moreover, from the construction, it is easily seen that even when ∇t (x ) and ∇t (x )
1 0 2 0
are linearly dependent, ∇T(x ) = ∇t (x ) − ∇t (x ) is a non-zero vector. Therefore,
0 1 0 2 0
taking a smaller open neighborhood U of x if necessary, it follows that T−1(0)∩ U is
0 0 0
an n-dimensional submanifold of Rn+1. Therefore, Proposition 3 follows. (cid:3)
e e
Propositions 2, 3 clearly yield the following:
Corollary 1. Let φ : Sn → Rn+1 be a C∞ embedding. Then, the union Caust(φ)∪Sym(φ)
is a subset of Lebesgue measure zero in Rn+1.
3. Proof of Theorem 1
Let γ : Sn → R be a C∞ convex integrand, and let V be a neighborhood of γ in
+
C∞ (Sn,R ). It is sufficient to show that V ∩S∞(Sn,R ) 6= ∅. In order to construct an
conv + +
element of V ∩S∞(Sn,R ), we consider the C∞ embedding φ : Sn → Rn+1−{0} defined
+
as follows:
1
φ(θ) = θ, .
(cid:18) γ(−θ)(cid:19)
4 E.B. BATISTA,H. HAN,AND T. NISHIMURA
Let W be the convex hull of φ(Sn). Then, since γ is a convex integrand, it follows that
(∗) φ(Sn) = ∂W,
where ∂W stands for the boundary of W.
Next, for any v ∈ int(W) consider the parallel translation T : Rn+1 → Rn+1 defined by
v
T (x) = x−v, where int(W) means the set consisting of interior points of W. Moreover,
v
for any θ ∈ Sn set L = {(θ,r) ∈ Rn+1 − {0}|r ∈ R } and for any v ∈ int(W) define
θ +
γ˜ : Sn → R as follows.
v +
(θ,γ˜ (θ)) = T (∂W)∩L .
v v θ
Notice that, by (∗) and v ∈ int(W), γ is a well-defined function. Notice also that
v
graph(γ˜ ) = T (∂W). By (∗) and v ∈ int(W) again, it follows that ||φ(θ)− v|| > 0 for
v v
any θ ∈ Sn. Thus, it follows that the maepping h : Sn → Sn defined by
v
φ(θ)−v
h (θ) =
v
||φ(θ)−v||
is a C∞ diffeomorphism and the following holds:
(γ˜ ◦h )(θ) = ||φ(θ)−v||.
v v
Let H : R → R be the C∞ diffeomorphism defined by H(X) = 1X2. Then, we have
+ + 2
the following:
1
F(v,θ) = ||φ(θ)−v||2 = (H ◦γ˜ ◦h )(θ).
v v
2
Hence, we have
Caust(φ) = {v : ∃θ : ∇(H ◦γ˜ ◦h )(θ) = 0 and det(Hess(H ◦γ˜ ◦h )(θ)) = 0}
v v v v
= {v : ∃θ : ∇(γ˜ ◦h )(θ) = 0 and det(Hess(γ˜ ◦h )(θ)) = 0}
v v v v
and
Sym(φ)
= {v : ∃θ 6= θ : ∇(H ◦γ˜ ◦h )(θ ) = ∇(H ◦γ˜ ◦h )(θ ) = 0
1 2 v v 1 v v 2
and (H ◦γ˜ ◦h )(θ ) = (H ◦γ˜ ◦h )(θ )}
v v 1 v v 2
= {v : ∃θ 6= θ : ∇(γ˜ ◦h )(θ ) = ∇(γ˜ ◦h )(θ ) = 0 and (γ˜ ◦h )(θ ) = (γ˜ ◦h )(θ )}.
1 2 v v 1 v v 2 v v 1 v v 2
For any r ∈ R , let B(0,r) be the open disk with radius r centered at 0. Then, by
+
Corollary 1, for any sufficiently small ε > 0 there exists a point v ∈ B(0,ε) such that
γ ◦h is stable. This implies that there exists a sequence {v ∈ int(W)} converging
v v n n=1,2,...
to the origin such that γ is stable for any n ∈ N.
vn
e For any v ∈ int(W), define the convex integrand γ : Sn → R as follows:
v +
e
1
γ (θ) = .
v
γ (−θ)
v
Since Sn is compact, the mapping Φ : B(0,eε) → Cc∞onv(Sn,R+) defined by Φ(v) = γv
is continuous. Since γ = γ, it follows that if n is sufficiently large, then the convex
0
integrand γ must be inside the given neighborhood V of γ. ✷
vn
STABILITY OF C∞ CONVEX INTEGRANDS 5
4. Application of the proof of Theorem 1
Theorem 1 guarantees that C∞ (Sn,R ) has sufficiently many stable functions. As in
conv +
[21], C∞ convex integrands γ having only non-degenerate critical points can be a useful
tool to investigate inv(graph(γ)). For instance, we have the following:
Proposition 4. Let n be an integer satisfying n ≥ 2. Let γ : Sn → R be a C∞ convex
+
integrand such that any critical point is non-degenerate and the index of γ at any critical
point is zero or n. Then, γ must be stable.
Notice that Proposition 4 does not hold in the case n = 1.
Proof. The following lemma is needed.
Lemma 4.1. Let n be an integer satisfying n ≥ 2. Let γ : Sn → R be a convex integrand
+
having only non-degenerate critical points such that the index of γ at any critical point is
zero or n. Then, γ has only two critical points, the index is zero at one point and it is n
at another point.
Lemma 4.1 can beproved by using the Morse inequalities as follows. Forany non-negative
integer λ, denote by C the number of critical points of index λ and by R the λ-th Betti
λ λ
number of Sn. Set
S = R −R +···±R .
λ λ λ−1 0
Then, the following inequalities, called the Morse inequalities, hold ([21]):
(i ) S ≤ C −C +···±C .
λ λ λ λ−1 0
The inequality (i ) implies 1 ≤ C . Since n ≥ 2, the inequality (i ) implies 0−1 ≤ 0−C ,
0 0 1 0
which is equivalent to C ≤ 1. Thus, we have C = 1. In the case λ > n, the inequalities
0 0
(i ),(i ) imply the following:
λ λ+1
n n
(−1)λR = (−1)λC .
λ λ
Xλ=0 Xλ=0
By using this equality, it is easily seen that C = 1; and thus the proof of Lemma 4.1
n
completes.
Now we prove Proposition 4. It is sufficient to show the following inclusion:
Sym(φ)∩int(W) ⊂ Caust(φ)∩int(W).
Here, φ : Sn → Rn+1 is the C∞ embedding defined by φ(θ) = θ, 1 , Sym(φ) is the
γ(−θ)
(cid:16) (cid:17)
symmetry set of φ, Caust(φ) is the caustic of φ and int(W) is the set consisting of interior
points of the convex hull of inv(graph(γ)). Suppose that the following set is non-empty.
Sym(φ)∩int(W)−Caust(φ)∩int(W).
Then, take one element v of this set. For the v, as in Section 3, we construct a C∞ convex
integrand H ◦ γ ◦ h . Since v is outside Caust(φ) ∩ int(W), it follows that all critical
v v
point of H ◦ γ ◦ h are non-degenerate. By Lemma 4.1, the function H ◦ γ ◦ h has
v v v v
only two critical points, the index is zero at one point and it is n at another point. Let
θ (resp., θ ) be the critical point with index zero (resp., index n). Then, it follows that
1 2
H◦γ ◦h (θ ) is the minimal value and H◦γ ◦h (θ ) is the maximal value. On the other
v v 1 v v 2
hand, since v is inside Sym(φ)∩int(W), there exist two distinct critical points θ ,θ ∈ Sn
1 2
such that H◦γ ◦h (θ ) = H◦γ ◦h (θ ). Since there are no critical points of H◦γ ◦h
v v 1 v v 2 e e v v
e e
6 E.B. BATISTA,H. HAN,AND T. NISHIMURA
except for θ ,θ , it follows that {θ ,θ } = {θ ,θ }. Therefore, H ◦ γ ◦ h must be a
1 2 1 2 1 2 v v
constant function, and this implies that φ(Sn) is a sphere centered at v. Hence, it follows
e e
that Caust(φ) = Sym(φ) = {v}, which contradicts the assumption that v 6∈ Caust(φ).
Therefore, the following inclusion holds:
Sym(φ)∩int(W) ⊂ Caust(φ)∩int(W).
(cid:3)
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STABILITY OF C∞ CONVEX INTEGRANDS 7
Research Institute of Environment and Information Sciences, Yokohama National
University, Yokohama 240-8501, Japan
E-mail address: [email protected]
Graduate School of Environment and Information Sciences,Yokohama National Uni-
versity, Yokohama 240-8501, Japan
E-mail address: [email protected]
Research Institute of Environment and Information Sciences, Yokohama National
University, Yokohama 240-8501, Japan
E-mail address: [email protected]