We study the growth of harmonic functions on complete Riemann-ian manifolds
where the extrinsic diameter of geodesic spheres is sublinear. It is an
generalization of a result of A. Kazue. We also get a Cheng and Yau estimates
for the gradient of harmonic functions

Carron, Gilles

Annales mathématiques Blaise Pascal

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arXiv

International audienceWe study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kazue. We also get a Cheng and Yau estimates for the gradient of harmonic functions.On étudie la croissance des fonctions harmoniques sur les variétés riemanniennes complètes dont le diamètre des grandes sphères géodésiques croit sous linéairement. Il s'agit de généralisation de travaux de A. Kazue. Nous obtenons aussi une estimée de type Cheng-Yau pour le gradient des fonctions harmoniques

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HARMONIC FUNCTIONS ON MANIFOLDS WHOSE LARGE SPHERE ARE SMALL

Gilles Carron

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Harmonic functions on Manifolds whose large spheres are small.

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HARMONIC FUNCTIONS ON MANIFOLDS WHOSELARGE SPHERE ARE SMALLGilles CarronTo cite this version:Gilles Carron. HARMONIC FUNCTIONS ONMANIFOLDSWHOSE LARGE SPHERE ARESMALL. 2015. <hal-01132922>HAL Id: hal-01132922https://hal.archives-ouvertes.fr/hal-01132922Submitted on 18 Mar 2015HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.HARMONIC FUNCTIONS ON MANIFOLDS WHOSE LARGE SPHEREARE SMALL .GILLES CARRONABSTRACT. We study the growth of harmonic functions on complete Riemann-ian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It isan generalization of a result of A. Kazue. We also get a Cheng and Yau estimatesfor the gradient of harmonic functions.RE´SUME´ : On e´tudie la croissance des fonctions harmoniques sur les varie´te´sriemanniennes comple`tes dont le diame`tre des grandes sphe`des ge´ode´siques croitsous line´airement. Il s’agit de ge´ne´ralisation de travaux de A. Kazue. Nousobtenons aussi une estime´e de type Cheng-Yau pour le gradient des fonctionsharmoniques.1. INTRODUCTIONWhen (M,g) is a complete Riemannian manifold with non negative Ricci curva-ture, S-Y. Cheng and S-T. Yau have proven that any harmonic function h : M → Rsatisfies the gradient estimate [4] :supz∈B(x,R)|dh|(z) ≤C(n)Rsupz∈B(x,2R)|h(z)|.This result implies that such a manifold can not carry non constant harmonic func-tion h : M → R with sublinear growth :|h(x)| = o(d(o, x)), d(o, x) → +∞ .A celebrated conjecture of S-T. Yau predicted the finite dimensionality of the spaceof harmonic functions with polynomial growth on a complete Riemannian mani-fold with non negative Ricci curvature :Hν(M,g) ={h ∈ C2(M) , ∆gh = 0, |h(x)| = O(dν(o, x))}.This conjecture has been proven by T. Colding and B. Minicozzi in a much moregeneral setting.We say that a complete Riemannian manifold (Mn, g) satisfies the doublingcondition if there is a constant ϑ such that for any x ∈M and radius R > 0 :volB(x, 2R) ≤ ϑ volB(x,R).If B ⊂M is a geodesic ball, we will use the notation r(B) for the radius of B andκB for the ball concentric to B and with radius κr(B). And if f is an integrablefunction on a subset Ω ⊂M , we will note fΩ its mean over Ω:fΩ =1vol Ω∫Ωf.12 GILLES CARRONWe say that a complete Riemannian manifold (Mn, g) satisfies the scale (L2)Poincare´ inequality if there is a constant µ such that for any ball B ⊂ M and anyfunction ϕ ∈ C1(2B):‖ϕ− ϕB‖2L2(B) ≤ µ r2(B)‖dϕ‖2L2(2B) .Theorem. [5] If (M,g) is a complete Riemannian manifold that is doubling andthat satisfies the scale Poincare´ inequality then for any ν, the space of harmonicfunction of polynomial growth of order ν has finite dimension:dimHν(M,g) < +∞.It is well known that a complete Riemannian manifold with non negative Riccicurvature is doubling and satisfies the scale Poincare´ inequality, hence the Yau’sconjecture is true.The proof is quantitative and gives a precise estimation of the dimension ofdimHν(M,g). In fact, the condition on the Poincare´ inequality can be weakenedand the result holds on a doubling manifold (M,g) that satisfies the mean valueestimation [6, 10] : for any harmonic function defined over a geodesic ball 3B :supx∈B|h(x)| ≤Cvol 2B∫2B|h|.An example of Riemannian manifold satisfying the above condition are Rie-mannian manifold (M,g) that outside a compact set (M,g) is isometric to thewarped product([1,∞)× Σ, (dr)2 + r2γh)where (Σ, h) is a closed connected manifold and γ ∈ (0, 1]. But when γ ∈ (0, 1),a direct analysis, separation of variables, shows that any harmonic function h sat-isfying for some  > 0:h(x) = O(eCr1−γ−)is necessary constant. In particular, a harmonic function with polynomial growth isconstant. In [8, 9], A. Kasue has shown that this was a general result for manifoldwhose Ricci curvature satisfies a quadratic decay lower bound and whose geodesicspheres have sublinear growth (see also [11] for a related results):Theorem. If (M,g) is complete Riemannian manifold with a based point o whoseRicci curvature satisfies a quadratic decay lower bound:Ricci ≥ −κ2d2(o, x)g ,and whose geodeosic sphere have sublinear growth:diam ∂B(o,R) = o(R) , R→ +∞then any harmonic function with polynomial growth is constant.Following A. Grigor’yan and L. Saloff-Coste [7], we say that a ball B(x, r) isremote (from a fixed point o) if3r ≤ d(o, x).HARMONIC FUNCTIONS ON MANIFOLDS WHOSE LARGE SPHERE ARE SMALL . 3Our first main result is a refinement of A. Kasue’s result when the hypothesis of theRicci curvature is replaced by a scale Poincare´ inequality for remote ball : Thereis a constant µ such that all remote balls B = B(x, r) satisfy a scale Poincare´inequality :∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µr2‖dϕ‖2L2(2B)Theorem A. Let (M,g) be a complete Riemannian manifold whose remote ballssatisfy the scale Poincare´ inequality and assume that geodesic spheres have sub-linear growth:diam ∂B(o,R) = o(R) , R→ +∞.If h : M → R is a harmonic function such that for IR :=∫B(o,R) h2 :lim infR→+∞log(IR)diam ∂B(o,R)R= 0then h is constant.For instance, on such a manifold, a harmonic function h : M → R satisfying :|h(x)| ≤ Cd(o, x)ν (volB(o, d(o, x)))−12is constant. Moreover if the diameter of geodesic sphere satisfiesdiam ∂B(o,R) ≤ CRγ ,for some γ ∈ (0, 1) then if h : M → R is a harmonic function such that for somepositive constant C and :|h(x)| ≤ Ced(o,x)1−γ−volB(o, d(o, x))then h is constant.A by product of the proof will imply that on the class of manifold considered byA. Kasue, the doubling condition implies an estimate a` la Cheng-Yau for for thegradient of harmonic function:Theorem B. Let (Mn, g) be a complete Riemannian manifold that is doubling andwhose Ricci curvature satisfies a quadratic decay lower bound. Assume that thediameter of geodesic sphere has a sublinear growthdiam ∂B(o,R) = supx,y∈∂B(o,R)d(x, y) = o(R) ,then there is a constant C such that for any geodesic ball B ⊂ M and anyharmonic function h : 3B → Rsupx∈B|dh|2(x) ≤Cvol 2B∫2B|dh|2.This result has consequences for the boundness of the Riesz transform. When(Mn, g) is a complete Riemannian manifold with infinite volume, the Green for-mula and the spectral theorem yield the equality:∀f ∈ C∞0 (M) ,∫M|df |2g dvolg = 〈∆f, f〉L2 =∫M∣∣∣∆ 12 f ∣∣∣2 dvolg .4 GILLES CARRONHence the Riesz transformR := d∆−12 : L2(M)→ L2(T ∗M)is a bounded operator. It is well known [12] that on a Euclidean space, the Riesztransform has a bounded extensionR : Lp(Rn)→ Lp(T ∗Rn) for every p ∈ (1,+∞).Also according to D. Bakry, the same is true on manifolds with non-negative Riccicurvature [2]. As it was noticed in [3, section 5], in the setting of the Theorem B,the analysis of A. Grigor’yan and L. Saloff-Coste [7] implies a scale L1-Poincare´inequality: there is a constant C such that any balls B = B(x, r) satisfies :∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖L1(B) ≤ Cr2‖dϕ‖L1(2B) .And according to the analysis of P. Auscher and T. Coulhon [1] (see also the ex-planations in [3, section 5]), the Theorem B implies :Corollary C. Under the assumption of Theorem B, the Riesz transform is boundedon Lp for every p ∈ (1,+∞).Acknowledgements. I thank Hans-Joachim Hein : this project had begun by avery fruitful discussion where we proved together the key lemma (2.1). I’m par-tially supported by the grants ACG: ANR-10-BLAN 0105 and GTO : ANR-12-BS01-0004.2. ABSENCE OF HARMONIC FUNCTIONSRecall that when (M,g) is a complete Riemannian manifold and o ∈ M , wesay that a geodesic ball B(x, r) is remote (from o) if3r ≤ d(o, x).We define ρ the radius function by ρ(t) = infx∈∂B(o,t)maxy∈∂B(o,t)d(x, y), we haveρ(t) ≤ diam ∂B(o, t) ≤ 2ρ(t).2.1. An inequality.Lemma 2.1. Let (M,g) be a complete Riemannian manifold whose all remoteballs B = B(x, r) satisfy a scale Poincare´ inequality :∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µ r2(B)‖dϕ‖2L2(2B) .Then there are constants C > 0 and κ ∈ (0, 1) depending only on µ such that if∀r ∈ [R, 2R] : ρ(r) ≤ εr with ε ∈ (0, 1/12)and if h is a harmonic function on B(o, 2R) then∫B(o,R)|dh|2 ≤ C κ1ε∫B(o,2R)|dh|2 .HARMONIC FUNCTIONS ON MANIFOLDS WHOSE LARGE SPHERE ARE SMALL . 5Proof. Let r ∈ [R + 4εR, 2R − 4εR], our hypothesis implies that there is somex ∈ ∂B(o, r) such thatB(o, r + εR) \B(o, r) ⊂ B(x, εR+ εr).Let h : B(o, 2R) → R be a harmonic function and c ∈ R a real number. We usethe Lipschitz function :χ(x) =1 on B(o, r)r+εR−d(o,x)εR on B(o, r + εR) \B(o, r)0 outside B(o, r + εR)Then integrating by part and using the fact that h is harmonic we get∫Mχ2|d(h− c)|2 + 2χ(h− c)〈dχ, d(h− c)〉 =∫M〈d((h− c)χ2), d(h− c)〉 = 0So that we have :∫M|d(χ(h − c))|2 =∫Mχ2|d(h− c)|2 + 2χ(h− c)〈dχ, d(h − c)〉+ (h− c)2|dχ|2=∫B(o,r+εR)(h− c)2|dχ|2 ,and hence∫B(o,r)|dh|2 ≤∫B(o,r+εR)|d(χ(h − c))|2 =∫B(o,r+εR)(h− c)2|dχ|2≤1ε2R2∫B(o,r+εR)\B(o,r)(h− c)2≤1ε2R2∫B(x,εR+εr)(h− c)2.The hypothesis that ε ≤ 1/12 implies that the ball B(x, εR+ εr) is remote, henceif we choosec = hB(x,ε(R+r)) =1volB(x, , ε(R + r))∫B(x,,ε(R+r))hthen the Poincare´ inequality and the fact that r +R ≤ 3R imply :∫B(o,r)|dh|2 ≤ 9µ∫B(x,6εR)|dh|2.But we have :B(x, 6εR) ⊂ B(o, r + 6εR) \B(o, r − 6εR) ,hence we get ∫B(o,r−6εR)|dh|2 ≤ 9µ∫B(o,r+6εR)\B(o,r−6εR)|dh|2.6 GILLES CARRONAnd for all r ∈ [R,R− 12εR] we get :∫B(o,r)|dh|2 ≤9µ1 + 9µ∫B(o,r+12εR)|dh|2.We iterate this inequality and get∫B(o,R)|dh|2 ≤(9µ1 + 9µ)N ∫B(o,2R)|dh|2provide that N12εR ≤ R ; hence the result with C = 1 + 19µ andκ =(9µ1 + 9µ) 112.2.2. Harmonic function with polynomial growth. We can now prove the fol-lowing extension of Kasue’s results :Theorem 2.2. Let (M,g) be a complete Riemannian manifold whose all remoteballs B = B(x, r) satisfy a scale Poincare´ inequality :∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µr2(B)‖dϕ‖2L2(2B)Assume that balls anchored at o have polynomial growth :volB(o,R) ≤ CRµand that geodesic spheres have sublinear diameter growth :limt→+∞ρ(t)t= 0then any harmonic function on (M,g) with polynomial growth is constant.Proof. Let h : M → R be a harmonic function with polynomial growth :h(x) ≤ C(1 + d(o, x))ν .We will definedER =∫B(o,R)|dh|2 and (r) = supt≥rρ(t)t.We remark first that using the cut off function ξ defined byξ(x) =1 on B(o,R)2R−d(o,x)R on B(o, 2R) \B(o,R)0 outside B(o, 2R)We obtain(1) ER ≤∫B(o,2R)|d(ξh)|2 =∫B(o,2R)|h|2|dξ|2 ≤ CR2ν+µ−2.HARMONIC FUNCTIONS ON MANIFOLDS WHOSE LARGE SPHERE ARE SMALL . 7If we iterate the inequality obtained in Lemma 2.1, we get for all R such that(R) ≤ 1/12 :ER ≤ C`κ∑`−1j=01(2jR) E2`R .Using the estimation (1), we get(2) ER ≤ C(R)e`(log κ`∑`−1j=01(2jR)+log(2)(2ν+µ−2)+logC).But the Cesaro theorem convergence implies that :lim`→+∞1``−1∑j=01 (2jR)= +∞hence if we let ` → +∞ in the inequality (2) we get ER = 0 and this for allsufficiently large R, hence h is constant. 2.3. Extension. A slight variation of the arguments yields the following exten-sion:Theorem 2.3. Let (M,g) be a complete Riemannian manifold whose all remoteballs B = B(x, r) satisfies a scale Poincare´ inequality :∀ϕ ∈ C1(2B) : ‖ϕ− ϕB‖2L2(B) ≤ µr2(B)‖dϕ‖2L2(2B)Assume that the geodesic spheres have sublinear diameter growth :limt→+∞ρ(t)t= 0 and let (r) = supt≥rρ(t)t.Let h : M → R be a harmonic function and assume that IR =∫B(o,R)h2 satisfylog I(R) = o(∫ R/41dtt(t))then h is constant.Proof. Indeed, the above argumentation shows that if R is large enough thenER ≤M(`,R)I(2`+1R) 4−`R−2wherelog(M(`,R)) = log(C`κ∑`−1j=01(2jR))= ` logC + log κ`−1∑j=01 (2jR) .But`−1∑j=01 (2jR)≥1log 2`−1∑j=0∫ 2jR2j−1Rdtt (t)≥1log 2∫ 2`−1RR/2dtt (t).8 GILLES CARRONHence we get the inequality :logER ≤ log I(2`+1R)− ` log(4) + ` logC +log κlog 2∫ 2`−1RR/2dtt (t)− 2 logR.It is then easy to conclude. 3. LIPSCHITZ REGULARITY OF HARMONIC FUNCTIONSWe are going to prove that a Lipschitz regularity for harmonic function analo-gous to the the Cheng-Yau gradient inequality :Theorem 3.1. Let (Mn, g) be a complete Riemannian manifold that satisfy thedoubling condition : there is a constant ϑ such that for any x ∈ M and radiusR > 0 :volB(x, 2R) ≤ ϑ volB(x,R)and assume moreover that the Ricci curvature satisfies a quadratic decay lowerboundRicci ≥ −κ2r2(x)g ,where for a fixed point o ∈M : r(x) := d(o, x).Assume that the diameter of geodesic sphere growth slowlydiam ∂B(o,R) = supx,y∈∂B(o,R)d(x, y) = o(R)then there is a constant C such that for any geodesic ball B ⊂ M and any har-monic function h : 3B → Rsupx∈B|dh|2(x) ≤Cvol 2B∫2B|dh|2.Proof. According to [3, Proposition 5.3], we need only to show that there is aconstant C such that if R > 0 and if h : B(o, 2R) → R is a harmonic functionthen for any s ≤ σ ≤ R :(3)1volB(o, s)∫B(o,s)|dh|2 ≤CvolB(o, σ)∫B(o,σ)|dh|2.According to the Lemma 2.1, for all η > 0, there is a R0 > 0 such that for allR ≥ R0, then ∫B(o,R)|dh|2 ≤ η∫B(o,2R)|dh|2.Hence for all R ≥ R0 :1volB(o,R)∫B(o,R)|dh|2 ≤ η ϑ1volB(o, 2R)∫B(o,2R)|dh|2.Choose η = ϑ−1, then we get that for all R0 ≤ s ≤ σ ≤ R :1volB(o, s)∫B(o,s)|dh|2 ≤ϑvolB(o, σ)∫B(o,σ)|dh|2.HARMONIC FUNCTIONS ON MANIFOLDS WHOSE LARGE SPHERE ARE SMALL . 9The Ricci curvature being bounded on B(o, 3R0), the Cheng and Yau gradientestimate yields a constant B such that for all x ∈ B(o,R0) :|dh|2(x) ≤BvolB(o, 2R0)∫B(o,2R0)|dh|2Hence the estimate (3) holds with C = max{B,ϑ}. REFERENCES[1] P. Auscher, T. Coulhon, Riesz transform on manifolds and Poincare´ inequalities, Ann. Sc.Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 3, 531–555.[2] D. Bakry : Etude des Transformations de Riesz dans les varie´te´s Riemanniennes a` courburede Ricci minore´e. Se´minaire de probabilite´s XIX, 351:145–174, 1983-1984. Lecture Notesin Mathematics, 1123.[3] G. Carron, Riesz transform on manifolds with quadratic curvature decay , preprintarXiv:1403.6278.[4] S.Y. Cheng, S.T. Yau : Differential equations on Riemannian manifolds and their geometricapplications, Comm. Pure Appl. Math. 28 (1975), 333–354.[5] T-H. Colding, W-P. Minicozzi : Harmonic functions on manifolds. Ann. of Math. (2) 146(1997), no. 3, 725–747.[6] T-H. Colding, W-P. Minicozzi : Liouville theorems for harmonic sections and applications.Commun. Pure Appl. Math.51 (1998), 113–138.[7] A. Grigor’yan, L. Saloff-Coste : Stability results for Harnack inequalities, Ann. Inst. Fourier(Grenoble) 55 (2005), 825–890.[8] A. Kasue : Harmonic functions of polynomial growth on complete Riemannian manifoldsI, Proc. Symp. Pure Math., Vol. 54, part II, (1993), 281–290.[9] A. Kasue, Harmonic functions of polynomial growth on complete Riemannian manifoldsII,J. Math. Soc. Japan, 47 (1995), 37–65.[10] P. Li : Harmonic sections of polynomial growth. Math. Res. Lett. 4(1) (1997) 35–44.[11] C. Sormani : Harmonic functions on manifolds with nonnegative Ricci curvature and linearvolume growth. Pacific J. Math. 192 (2000), no. 1, 183–189.[12] E.M. Stein : Singular Integrals and Differentiability of Functions, Princeton UniversityPress, Princeton (1970).[13] S.T. Yau : Some function-theoretic properties of complete Riemannian manifold and theirapplications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670.LABORATOIRE DE MATHE´MATIQUES JEAN LERAY (UMR 6629), UNIVERSITE´ DE NANTES,2, RUE DE LA HOUSSINIE`RE, B.P. 92208, 44322 NANTES CEDEX 3, FRANCEE-mail address: Gilles.Carron@univ-nantes.fr

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Harmonic Functions On Manifolds Whose Large Sphere Are Small

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