On the Omori-Yau Maximum Principle and Geometric Applications

0.24 MB·English

by  Barnabe Pessoa LIma

1
ON THE OMORI-YAU MAXIMUM PRINCIPLE AND GEOMETRIC
APPLICATIONS
BARNABEPESSOALIMAANDLEANDRODEFREITASPESSOA
Abstract. We introduce a version of the Omori-Yau maximum principle which generalizes
theversionobtained byPigola-Rigoli-Setti[21]. Weapplyourmethodtoderiveanon-trivial
generalizationJorge-KoutrofiotisTheorem[15]forcylindricallyboundedsubmanifoldsdueto
2
Alias-Bessa-Montenegro [2], we extend results due to Alias-Dajczer [5], Alias-Bessa-Dajczer
1
[1]andAlias-Impera-Rigoli[6].
0
2
1. Introduction and Statement of Results
n
a H.Omori[17],studyingisometricimmersionsofminimalsubmanifoldsintoconesofRnproved
J
the following global version of the maximum principle for complete Riemannian manifolds with
9
sectional curvature bounded below.

] Theorem 1 (Omori). Let M be a complete Riemannian manifold with sectional curvature
G bounded below K >−Λ2. If u C2(M) with u∗ =sup u< then there exists a sequence of
D M ∈ M
points x M, depending on M and on u, such that
k
h. (1) ∈ lim u(x )=u∗, |gradu|(x )< 1, Hessu(x∞)(X,X)< 1 |X|2,
at k→ k k k k k ·
m for every X∈TxkM∞.
[ H. Omori’s maximum principle was refined by S. T. Yau in a series of papers [24], [25], [11]

(this later with S. Y. Cheng) and applied to find elegant solutions to various analytic-geometric
1
v problems on Riemannian manifolds. The version of the maximum principle Cheng-Yau proved
5 is the following variation of Theorem 1.
7
Theorem 2 (Cheng-Yau). Let M be a complete Riemannian manifold with Ricci curvature
6
1 bounded below RicM > −(n−1)·Λ2. Then for any u ∈ C2(M) with u∗ = supMu < , there
1. exists a sequence of points xk ∈M, depending on M and on u, such that ∞
0 (2) lim u(x )=u∗, |gradu|(x )< 1, u(x )< 1.
2 k→ k k k △ k k
1
H.Omori’sTheoremwasextendedbyC.Diasin[13]andCheng-Yau’sTheoremwasextended
: ∞
v by Chen-Xin in [12]. Recently, S. Pigola, M. Rigoli and A. Setti in their beautiful book [21]
Xi introduced the following important concept.
r Definition 1. The Omori-Yau maximum principle is said to hold on M if for any given u
a C2(M) with u∗ = sup u < , there exists a sequence of points x M, depending on M an∈d
M k ∈
on u, such that
∞
1 1
(3) lim u(x )=u∗, |gradu|(x )< , u(x )< .
k k k
k→ k △ k
12010 Mathematics Subjec∞t Classification: Primary53C42,;Secondary35B50
Date:February28,2008.
Key words and phrases. Omori-Yaumaximumprinciple.
Thefirstauthor waspartiallysupportedbyPROCAD-CAPES.
ThesecondauthorwaspartiallysupportedbyPICME-CAPES.
1
2 BARNABEPESSOALIMAANDLEANDRODEFREITASPESSOA
Likewise, the Omori-Yau maximum principle for the Hessian is said to hold on M if for any
given u C2(M) with u∗ = sup u < , there exists a sequence of points x M, depending
∈ M k ∈
on M and on u, such that
∞
1 1
(4) lim u(x )=u∗, |gradu|(x )< , Hessu(x )(X,X)< |X|2,
k k k
k→ k k ·
for every X∈TxkM∞.
This concept was a new point of view of the Omori-Yau maximum principle. That is, some
geometries do hold the Omori-Yau maximum principle whereas some does not. That raised
naturallythe questionofwhatare the geometriesthat holdthe Omori-Yaumaximumprinciple?
The Omori-Yau maximum principle was shown to hold in several geometric settings, see for
instance [7], [14], [16], [20], [23]. Regarding this problem, S. Pigola, M. Rigoli and A. Setti [21,
pp. 7–10] proved the a general class of Riemannian manifolds hold the Omori-Yau maximum
principle. They proved the following theorem.
Theorem3(Pigola-Rigoli-Setti). LetMbeaRiemannianmanifoldandassumethatthereexists
a non-negative function γ satisfying the following:
C1) γ(x) + , as x ;
→ →
C2) A>0 su∞chthat |gradγ|∞6A √γ, off a compact set;
∃ ·
C3) B>0 suchthat γ6B γG(√γ), off a compact set;
∃ △ ·
where G:[0,+ ) [0, ) is a smooth function satisfying
→ p
∞ ∞
+ ds tG(√t)
(5) G(0)>0, G′(t)>0, =+ , limsup <+ .
Z0 ∞ G(s) t→+ G(t)
∞ ∞
Then the Omori-Yau maximum principle hpolds on M. ∞
If instead of C3) we assume the following stronger hypothesis
C4) B>0 such that Hessγ(.,.)6B γG(√γ) .,. , off a compact set.
∃ h i
Then the Omori-Yau maximum principle for the Hessian holds on M.
p
Remark 1. An example of a smooth function G C ([0,+ )) satisfying (5) is given by
G(t)=t2Πℓ (log(j)(t))2 for t 1, where log(j) is t∈he j-∞th iterated logarithm and ℓ N.
j=1 ≫ ∞ ∈
Remark 2. Quoting Pigola-Rigoli-Setti, “The proof of Theorem 3 shows that one needs γ to be
C2 only in a neighborhood of x . This is the case that γ = ρ2 is the square of the Riemannian
k
distance from a fixed point x and x is not on the cut locus of x . The case that x is the cut
o k o k
locus of x can be dealt with a trick of Calabi [10] so that we may assume that γ is always C2 in
o
a neighborhood of x .”, [21, Remark 1.11].
k
Corollary 1 (Pigola-Rigoli-Setti). Let M be a complete Riemannian manifold with Ricci cur-
vature satisfying
Ric(x)>−B2 G(ρ(x)), for ρ(x) 1
· ≫
where G C ([0,+ )) satisfies (5), ρ(x) = distM(x0,x), B R. Then γ = ρ2 satisfies
∈ ∈
C1−C3. Ther∞efore the Omori-Yau maximum principle holds on M by Theorem 3. This shows
that Theorem 3 exten∞ds Theorem 2.
Inthis paperwe give anextensionof Pigola-Rigoli-Setti’sTheorem3. We provethe following
result.
ON THE OMORI-YAU MAXIMUM PRINCIPLE AND GEOMETRIC APPLICATIONS 3
Theorem 4 (MainTheorem). Let Mbea completeRiemannian manifold and assumethat there
exists a non-negative function γ satisfying:
h1) γ(x) + , as x ;
→ →
∞ ∞
h2) A>0 suchthat |gradγ|6A G(γ) γ ds +1 off a compact set;
∃ · 0 √G(s)
(cid:18)R (cid:19)
p
h3) B>0 suchthat γ6B G(γ) γ ds +1 off a compact set;
∃ △ · 0 √G(s)
(cid:18)R (cid:19)
where G:[0,+ ) [0, )pis a smooth function satisfying
→
∞ ∞
+ ds
(6) G(0)>0, G′(t)>0, =+ .
Z0 ∞ G(s)
∞
u(x) p t ds
Then if u C2(M) satisfies lim = 0, where ϕ(t)=log +1 , then there
∈ x→ ϕ(γ(x)) Z0 G(s) !
exists a sequence xk ∈M, k∈N∞such that p
1 1
(7) |gradu|(x )< , u(x )<
k k
k △ k
γ ds
If instead of h3) we assume Hessγ(.,.)6 G(γ) +1 .,. off a compact set,
Z0 G(s) !h i
1 p
thenHessu(x )(.,.)< .,. . Moreover,ifu C2(M)isbopundedaboveu∗ =sup u< then
k kh i ∈ M
u(x ) u∗.
k → ∞
This result above shouldbe comparedwith a fairly recentresult [22, Cor. A1.] also due to S.
Pigola, M. Rigoli, A. Setti improved Theorem 3 to more general elliptic operators.
2. Omori-Yau maximum principle
Proof of Theorem 4: We fix asequenceofpositiverealnumbers (εk)k∈N suchthat, εk 0
→
and
u(x)
consider now any function u C2(M) satisfying lim = 0, where
∈ x→ ϕ(γ(x))
t ds ∞
ϕ(t)=log +1 . Define
Z0 G(s) !
(8) p g (x)=u(x)−ε ϕ(γ(x))
k k
and observe that ϕ is C2(M), positive and satisfies
ϕ(t) + as t + .
→ →
By a direct computation we have
∞ ∞
−1
t ds
ϕ′(t) = G(t) +1 ,
" Z0 G(s) !#
p
p −2
t ds G′(t) t ds
ϕ′′(t) = − G(t) +1 +1 +1
" Z0 G(s) !# "2 (G(t)) Z0 G(s) ! #
p
p p p
4 BARNABEPESSOALIMAANDLEANDRODEFREITASPESSOA
and using the properties satisfied by G we conclude that
(9) ϕ′′(t)60.
It is clear that g attains its supremum at some point x M. This gives the desired sequence
k k
∈
x . It follows directly from definition of g that
k k
gradg (x)=gradu(x)−ε ϕ′(γ(x))gradγ(x).
k k
In particular, at the points x we obtain
k
(10) |gradu|(x )=ε ϕ′(γ(x ))|gradγ|(x ).
k k k k
Using h2) in the above equality we have
|gradu|(x )6ε .
k k
Computing Hessg (x)(v,v) we have
k
Hessg (x)(v,v) = Hessu(x)(v,v)−ε ϕ′(γ(x))Hessγ(x)(v,v)
k k
(11) −ε ϕ′′(γ(x)) gradγ(x),v 2
k
h i
> Hessu(x)(v,v)−ε ϕ′(γ(x))Hessγ(x)(v,v)
k
for all v T M. Using the fact that x is a maximum point of g , the hypothesis h4) and the
x k k
expressio∈n for ϕ′, we get
(12) Hessu(x )(v,v)6ε ϕ′(γ(x ))Hessγ(x )(v,v)6ε v,v .
k k k k k
h i
Finally, if assume that h3) holds, we obtain
∆u(x )6ε ϕ′(γ(x ))∆γ(x )6ε .
k k k k k
To finish the proof of Theorem 4 we need to show that if u∗ = sup u < then u(x ) u∗.
To do that, we follow Pigola-Rigoli-Setti closely in [21] and observMe that for any fixekd j→ N,
there is a y M such that ∞ ∈
∈
(13) u(y)>supu−1/2j.
Since g has a maximum at x we have
k k
γ(xk) ds γ(y) ds
u(x )−ε log +1 =g (x )>g (y)=u(y)−ε log +1 .
k k k k k k
Z0 G(s) ! Z0 G(s) !
Therefore, (using (13))p p
1 γ(y) ds
(14) u(x )>supu− −ε log +1 .
k k
2j Z0 G(s) !
Choosing k=k(j)=kj sufficiently large such that p
γ(y) ds 1
(15) ε log +1 < ,
kj Z0 G(s) ! 2j
it follows from (14) and (15) that p
1 1 1
(16) u(x )>supu− − =supu− .
kj 2j 2j j
Therefore lim u(x )=supu and this finishes the proof of Theorem 4.
j→+ kj M
∞
ON THE OMORI-YAU MAXIMUM PRINCIPLE AND GEOMETRIC APPLICATIONS 5
Remark 3. Let G and G be the classes of Riemannian manifolds satisfying respectively the
hypotheses of the Theorem 3 and Theorem 4. Then we have that G G. Hence, the Theorem 4
⊂
implies the Pigola-Rigoli-Setti’s Theorem (Thm. 3).
Proof. Given M G, observe that the hypothesis
∈
tG(√t)
limsup =D<+
t→+ (cid:20) G(t) (cid:21)
implies the existence of s0 R such th∞at ∞
∈
tG(√t)
(17) sup ,t>s0 <D+1.
(cid:12) G(t) (cid:13)
Thus for all t>s0 we have that
tG(√t)
(18) <D+1 tG(√t)<(D+1)G(t)
G(t) ⇔
whence tG(√t)< (D+1)G(t). In particular
q
p γ ds
A γG(√γ)<A (D+1)G(γ)6A (D+1) G(γ) +1 .
Z0 G(s) !
q p p p
Finally, we refer to [21, p.10] for a proof that any M G is a complepte manifold. (cid:3)
∈
Remark 4. Estimates placed the items h2), h3) e h4) can be exchanged for
k t ds t ds
(19) ln(j) +1 +1 +1 G(t),
Yj=1" Z0 G(s) ! # Z0 G(s) !
p
where ln(j) is the j-th iterated lopgarithm and k N. p
∈
Proof. Indeed,notethatthisestimateissimplythe inverseofthe firstderivativeoftheauxiliary
function ϕ, used in the statement of Main Theorem 4. Thus, we redefine the function ϕ by
t ds
ϕ(t)=ln(k+1) +1 .
Z0 G(s) !
Hence, we obtain by deriving p
−1
ϕ′(t)= k ln(j) t ds +1 +1 t ds +1 G(t)
Yj=1" Z0 G(s) ! # Z0 G(s) ! 
p
and  p p 
−2
ϕ′′(t) = − k ln(j) t ds +1 +1 t ds +1 G(t)
Yj=1" Z0 G(s) ! # Z0 G(s) !  ×
p
k−1 i pt ds −p1 t ds  −k−1
 ln(j) +1 +1 +1 G(t) +
×Yi=1Yj=1" Z0 G(s) ! # " Z0 G(s) ! #
p
k t ds p G′(t) tp ds
+ ln(j) +1 +1 +1 +1 
Yj=1" Z0 G(s) ! #"2 G(t) Z0 G(s) ! #
6 0. p p p 
6 BARNABEPESSOALIMAANDLEANDRODEFREITASPESSOA
Thereforethe function ϕ satisfies the conditions necessaryto provethe Main Theorem4. (cid:3)
Corollary 2. Let M be a complete, noncompact, Riemannian manifold with Ricci curvature
satisfying
2 2
k ρ(x) ds ρ(x) ds
Ric(x)>−B2 ln(j) +1 +1 +1 G(ρ(x)),
Yj=1" Z0 G(s) ! # Z0 G(s) !
for ρ(x) 1 where G C ([0,+ p)) satisfies (6), ρ(x) = disptM(x0,x), B R. Then γ = ρ
≫ ∈ ∈
satisfies h1−h3. Therefore∞the Omori-Yau maximum principle holds on M for the Laplacian by
Main Theorem 4. Similarly, if we∞assume that the radial sectional curvature satisfies the above
inequality, then the Omori-Yau maximum principle holds on M for the Hessian.
3. Weighted Riemannian manifolds
Aweightedmanifold(M,g,µ ),shortlydenotedby(M,µ ),isaRiemannianmanifold(M,g)
f f
endowedwithameasureµ =e−fν,wheref: M Risasmoothfunctionandν=√detgdx1...xn
f
→
is the Riemannian density. The associated Laplace-Betramioperator is defined by
f
△
:=efdiv(e−fgrad).
f
△
It is natural to extend the results above to the weighted Laplacian. A. Borbely, [8] [9] proved
a nice extension of Pigola-Rigoli-Setti’s version [21] of the Omori-Yau maximum principle for
the Laplacian. Borbely’s version has been extended to the weighted Laplacian or even to more
general operators by many authors. Some in the weak form of the maximum principle others in
the strong form of the maximum principle. For instance, Bessa, Pigola and Setti in [7, Thm 9],
Pigola Rigoli and Setti [22], Mari, Rigoli and Setti in [19], by Pigola, Rigoli, Rimoldi and Setti
in [20] and by Mastrolia, Rigoli and Rimoldi in [18]. For the Laplace operator we can resume
what they proved as
Theorem 5 (Borb´ely, Bessa, Mari, Mastrolia, Pigola, Rigoli, Rimoldi, Setti). Let (M,µ ) be a
f
complete weighted manifold and assume that there exists a non-negative C2-function γ satisfying
the following conditions.
a. γ(x) + as x .
→ →
b. A>0 su∞ch that |gra∞dγ|<A off a compact set.
∃
c. B> such that γ6B G(γ) off a compact set.
f
∃ △ ·
Where G C ([0, )) satisfying
∈
(20) ∞ G(0)>0, G′(t)>0 in [0, ), G(t)−1 L1([0, ).
∞
6∈
Then the Omori-Yau maximum principle for holds on M.
f∞ ∞
△
The main result of this section is the following extension of Theorem 5.
Theorem 6. Let (M,µ ) be a complete weighted manifold and assume that there exists a non-
f
negative C2(M)-function γ satisfying the following conditions:
a. γ(x) + as x ;
→ →
∞ ∞
b. A>0 such that |gradγ|<A G(γ) γ ds +1 off a compact set;
∃ · 0 √G(γ)
(cid:18)R (cid:19)
p
ON THE OMORI-YAU MAXIMUM PRINCIPLE AND GEOMETRIC APPLICATIONS 7
c. B> such that γ6B G(γ) γ ds +1 off a compact set;
∃ △f · 0 √G(γ)
(cid:18)R (cid:19)
where G C ([0, )) satipsfies
∈
(21) G(0∞)>0, G′(t)>0 in [0, ), G(t)−1/2 L1([0, ).
∞
6∈
Then the Omori-Yau maximum principle for holds on M.
△f∞ ∞
Remark 5. Replacing the bound γ < A G(r) in Theorem 5, with G satisfying (20), by
f
△ ·
γ6B G˜(γ) γ ds +1 ,withG˜ satisfying (21)doesnotamounttoaweakercondition.
△f · 0 √G˜(γ)
q (cid:18)R (cid:19)
In fact, if G˜ satisfies (21) then G(r)= G˜ satisfies (20).
p u(x)
Remark 6. We remark that if u C2(M) with lim = 0, then there exist a sequence
∈ x→ ϕ(γ(x))
1
x M, k N such that u(x )< . ∞
k f k
∈ ∈ △ k
Proof. The proof follows closely the proof of the Main Theorem 4. We need only to adapt the
part of the proof that treats with the to . As in there we defined the functions sequence
f
△ △
g and observe that
k
ϕ(γ)(x) = ϕ′(γ(x)) γ(x)+ϕ′′(γ(x))|gradγ|2(x).
f f
△ △
Note that in the maximum points x of the functions g , we have g (x )60. Thus,
k k f k k
△
0 > g (x )= u(x )−ε ϕ(γ(x ))
f k k f k k f k
△ △ △
= u(x )−ε ϕ′(γ(x ) γ(x )+ϕ′′(γ(x ))|gradγ(x )|2 ,
f k k k f k k k
△ △
which implies in (cid:2) (cid:3)
u(x ) 6 ε [ϕ′(γ(x )) γ(x )+ϕ′′(γ(x ))|gradγ(x )|2]
f k k k f k k k
△ △
6 ε ϕ′(γ(x )) γ(x ).
k k f k
△
In the last inequality we used that ϕ′′ 60. Then the we proceed as in Theorem 4 to finish the
proof. (cid:3)
4. Geometric Applications
Inabeautifulpaper[15],JorgeandKoutrofiotisappliedOmori’sTheorem(1)togivecurvature
estimates for bounded submanifolds with scalar curvature bounded below, extending various
non-immersability results. Their result was extended by Pigola, Rigoli and Setti in [21] as an
application of their generalized version of the Omori-Yau maximum principle.
Recently, L. Alias, G. P. Bessa and J. F. Montenegro in [2] proved a version of Jorge-
Koutrofiotis Theorem for cylindrically bounded submanifolds, recalling that an isometric im-
mersionϕ: M֒ N Rℓ issaidtobe cylindricallyboundedifϕ(M) B (r) Rℓ,whereB (r)
N N
→ × ⊂ ×
is a geodesic ball in N of radius r>0.
Theorem 7. [Alias-Bessa-Montenegro] Let M and N be complete Riemannian manifolds of di-
mensionmandn−ℓrespectively,satisfyingn+ℓ62m−1. Letϕ:Mm Nn−ℓ Rℓ beaisomet-
ric immersion with ϕ(M) B (r) Rℓ. Assume that the radial sectio→nal curv×ature Krad along
⊂ N × N
the radial geodesics issuing from p satisfies Krad 6b in B (r) and 0<r<min{inj (p),π/2√b},
N N N
where we replace π/2√b by + if b60. Suppose that the immersion ϕ is proper and
(22) sup α 6σ(t),
∞
ϕ−1(BN(r)×∂BRℓ(t))k k
8 BARNABEPESSOALIMAANDLEANDRODEFREITASPESSOA
where α is the second fundamental form of the immersion and σ : [0,+ ) R is a positive
function satisfying + 1/σ = + , then the sectional curvature of M has th→e following lower
0
∞
bound R ∞
∞
(23) supK >C2(r)+ inf K ,
M b N
M BN(r)
with
√bcot(√bt) if b>0 and 0<t<π/2√b
C (t)=1/t if b=0 and t>0
b 
√−bcoth(√−bt) if b<0 and t>0.

In this section our main result is the following generalization of Theorem 7. We prove the
following result.
Theorem 8. Let M and N be complete Riemannian manifolds of dimension m and n − ℓ
respectively, satisfying n + ℓ 6 2m − 1. Let ϕ : Mm Nn−ℓ Rℓ be a proper isometric
immersion with ϕ(M) B (r) Rℓ. Assume that the rad→ial section×al curvature Krad along the
⊂ N × N
radial geodesics issuing from p satisfies Krad 6 b in B (r) and 0 < r < min{inj (p),π/2√b},
N N N
where we replace π/2√b by + if b 6 0. Then the sectional curvature of M has the following
lower bound
∞
(24) supK >C2(r)+ inf K .
M b N
M BN(r)
Proof. Let g:N Rℓ R be given g(z,y)=φ (ρ (z)), where φ is given by
b N b
× →
1−cos(√bt) if b>0 and 0<t<π/2√b
(25) φ (t)= t2 if b=0 and t>0
b
 cosh(√−bt) if b<0 and t>0
and ρ (z) = dist (p,z).Consider f : M R, f = g ϕ and let π : N Rℓ N be the
N N N
projectiononthefactorN. Sinceπ (ϕ(M))→ B (r),we◦havethatf∗ =sup ×f6φ→(r)<+ .
Define for each k N, the functionNg :M ⊂R bNe given M b
k
∈ → ∞
(26) gk(x)=f(x)−εkψ(ρRℓ(y(x))),
where ψ:R [0,+ ) is given by ψ(t)=log(log(t+1)+1), ρRℓ(y)=distRℓ(0,y), εk 0+ as
→ →
k and y(x)=πRℓ(ϕ(x)).
→Sincetheimmersi∞onϕisproper,ifx inMthenϕ(x) inB (r) Rℓ,thusy(x)
N
inRℓ∞andψ(ρRℓ(y(x))) + . Therefo→regk reachitsmaxim→umatapoint×xk M. Thisf→orms
a sequence {x } M suc→h that g (x )=s∞up g . There are tw∞o cases to cons∈ider: ∞
k ⊂ ∞ k k M k
1. x in M as k + .
k
→ →
2. x stays in a bounded subset of M.
k
∞ ∞
Let us consider the case 1. i.e. x in M as k + . Since x is a point of maximum for
k k
→ →
g we have that Hess g (X,X)60 for all X T M. This implies that
k M k ∞ ∈ xk ∞
(27) HessMf(xk)(X,X)6εkHessMψ◦ρRℓ ◦y(xk)(X,X), X∈TxkM.
First we will compute the right hand side of (27). We have then
HessMψ◦ρRℓ ◦y(xk)(X,X) = HessN×Rℓψ◦ρRℓ ◦y(xk)(X,X)
(28)
+hgradN×Rℓψ◦ρRℓ ◦y(xk),αM(X,X)i,
where α is the second fundamental form of the immersion ϕ, see [15].
ON THE OMORI-YAU MAXIMUM PRINCIPLE AND GEOMETRIC APPLICATIONS 9
Setting yk =πRℓ(ϕ(xk)) and tk =ρRℓ(yk) we have
HessN×Rℓψ◦ρRℓ ◦y(xk)(X,X) = ψ′′(tk)|XRℓ|2+ψ′(tk)HessRℓρRℓ(yk)(X,X)
|XN|2
(29) = ψ′′(t )|XRℓ|2+
k t (t +1)(log(t +1)+1)
k k k
|X|2
6
t (t +1)(log(t +1)+1)
k k k
Since ψ′′ 60. Here XRℓ =dπRℓX and XN =dπNX, where πRℓ: N Rℓ Rℓ, πN: N Rℓ N
× → × →
are standard projections.
We also have
hgradN×Rℓψ◦ρRℓ ◦y(xk),αM(X,X)i = ψ′(tk)hgradρRℓ(yk),α(X,X)i
1
(30) 6 |α(X,X)|
(t +1)(log(t +1)+1)
k k
From (29) and (30) we have that
1+|α(X,X)|
(31) HessMψ◦ρRℓ ◦y(xk)(X,X) 6 (t +1)(log(t +1)+1)|X|2
k k
And from (27) and (31) we get
ε (1+|α(X,X)|)
(32) Hess f(x )(X,X) 6 k |X|2
M k (t +1)(log(t +1)+1)
k k
Now, we will compute the left hand side of (27).
(33) HessMf(xk)(X,X) = HessN×Rℓg(ϕ(x))(X,X)+hgradg,α(X,X)i
Recalling that f=g ϕ and g is given by g(z,y)=φ (ρ (z)), where φ is given by
b N b
◦
1−cos(√bt) if b>0 and 0<t<π/2√b
(34) φ (t)= t2 if b=0 and t>0
b
 cosh(√−bt) if b<0 and t>0.

and ρ (z)=dist (p,z). Let us consider an orthonormal basis
N N
∈TN ∈TRℓ
{gradρN,∂/∂θ1,...,∂/∂θn−ℓ−1,∂/∂γ1,...,∂/∂γℓ}
z }| { z }| {
for T (N Rℓ). Thus if X T M, |X|=1, we can decompose
ϕ(xk) × ∈ xk
n−ℓ−1 ℓ
X=a gradρ + b ∂/∂θ + c ∂/∂γ
N j j i i
· · ·
Xj=1 Xi=1
10 BARNABEPESSOALIMAANDLEANDRODEFREITASPESSOA
witha2+ n−ℓ−1b2+ ℓ c2 =1. Letting z =π (ϕ(x ))ands =ρ (z ). Havingsetthat
j=1 j i=1 i k N k k N k
we have thPat the first tePrm of the right hand side of (33)
n−ℓ−1
∂ ∂
HessN×Rℓg(ϕ(x))(X,X) = φb′′(sk)·a2+φb′(sk) b2j ·HessρN(zk)(∂θ ,∂θ )
Xj=1 j j
n−ℓ−1
> φ′′(s ) a2+φ′(s ) b2 C (s )
b k · b k j · b k
Xj=1
ℓ
= φ′′(s ) a2+(1−a2− c2) φ′(s ) C (s )
b k · i · b k · b k
Xi=1
≡0
ℓ
= (φ′′−C φ′)a2+(1− c2) φ′ C (s )
 b b· b i · b· b k
z }| { Xi=1
 
ℓ
= (1− c2) φ′(s ) C (s ).
i · b k · b k
Xi=1
Thus
ℓ
(35) HessN×Rℓg(ϕ(x))(X,X) >(1− c2i)·φb′(sk)·Cb(sk)·|X|2.
Xi=1
We used above two facts. The first was Hessρ (z )( ∂ , ∂ ) > C (s ) yielded by the
N k ∂θj ∂θj b k
Hessian Comparison Theorem, Thm. 9. Recall that the radial sectional curvature of N along
the geodesics issuing from the center of the ball B (r) is bounded above Krad 6 b, see the
N N
hypotheses of Theorem 8. We state the Hessian Comparison Theorem for sake of completeness.
The second fact is the φ satisfies the following equation φ′′(t)−C (t)φ′(t) = 0, C given
b b b b b
below in (37) .
Theorem 9 (Hessian Comparison Theorem). Let M be a Riemannian manifold and x0,x1
∈
M be such that there is a minimizing unit speed geodesic γ joining x0 and x1 and let ρ(x) =
dist(x0,x) be the distance function to x0. Let Kγ 6 b be the radial sectional curvatures of M
along γ. If b>0 assume ρ(x1)<π/2√b. Then, we have Hessρ(x)(γ˙,γ˙)=0 and
(36) Hessρ(x)(X,X)>C (ρ(x))|X|2
b
where X T M is perpendicular to γ˙(ρ(x)) and
x
∈
√bcot(√bt) if b>0 and 0<t<π/2√b
(37) C (t)= 1/t if b=0 and t>0
b
 √−bcoth(√−bt) if b<0 and t>0.

The second term of the right hand side of (33) is the following, if |X|=1.
gradg,α(X,X) = φ′(s ) gradρ (y ),α(X,X)
h i b k h N k i
(38)
> −φ′(s )|α(X,X)|
b k
Therefore from (33), (35), (38) we have that