YUXIANG LI DEPARTMENT OF MATHEMATICAL SCIENCES, TSINGHUA UNIVERSITY, BEIJING , P.R.CHINA.

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1 SOME REMARKS ON WILLMORE SURFACES EMBEDDED IN R 3 YUXIANG LI DEPARTMENT OF MATHEMATICAL SCIENCES, TSINGHUA UNIVERSITY, BEIJING , P.R.CHINA. YXLI@MATH.TSINGHUA.EDU.CN. Abstact. Let f : C R 3 be complete Willmoe immesion with Σ A f < +. We will show that if f is the it of an embedded suface sequence, then f is a plane. As an application, we pove that if Σ is a sequence of closed Willmoe suface embedded in R 3 with W (Σ ) < C, and if the confomal class of Σ conveges in the moduli space, then we can find a Möbius tansfomation σ, such that a subsequence of σ (Σ ) conveges smoothly. 1. Intoduction Let Σ be a Riemann suface and f : Σ R 3 be an embedding. We define the fist and second fundamental fom of f as follows: g = g ij dx i dx j = df df, and A = A ij dx i dx j = df dn. Let H = g ij A ij be the mean cuvatue, and K be the Causs cuvatue. It is well-nown that (1.1) H = Hn = g f. We say f is minimal if H = 0, and Willmoe if H satisfies the equation: (1.) g H + 1 ( H 4K)H = 0. Note that (1.) is the Eule-Langange equation of Willmoe functional [0]: W (f) = 1 H dµ g. 4 Now, we let f be an embedding fom C into R 3. We assume f is complete and noncompact, with C A < +. It is well-nown that when f is minimal, f must be a plane [16]. In this pape, we will show that such a esult is also tue when f is Willmoe: Theoem 1.1. Let f : C R 3 be a complete Willmoe embedding. If C A < +, then f(c) is a plane. Rema 1.. In [4], Chen and Lamm has poved that any Willmoe gaph ove R in R 3 must be a plane, wheneve it has finite A L. The autho is patially suppoted by NSFC Poject and NSFC Poject

2 Y. LI Luo and Sun poved that if the Willmoe functional of the Willmoe gaph is finite, then A L is finite [14]. Howeve, this is not tue fo an embedded Willmoe suface. Fo example, helicoids ae embedded minimal sufaces (W = 0), but have infinite A L. Next, we will show that Theoem 1.1 still holds if we eplace embedding with the it of an embedding sequence : Theoem 1.3. Let f : C R 3 be a confomal complete Willmoe immesion with C A < +. If thee exist R + and embedding φ : D R R 3, such that f conveges to f in C 1 (D R ) fo any R, then f(c) is a plane. As an application, we will pove the following: Theoem 1.4. Let Σ be a closed Willmoe suface embedded in R 3. We assume the genus is fixed and W (Σ ) < C. If the confomal class of Σ is contained in a compact subset of the moduli space, then we can find Möbius tansfomation σ, such that σ (Σ ) conveges smoothly. Rema 1.5. Let Σ be a Willmoe suface immesed in R 3 and R 4. Benad and Riviée [1] poved that if W (Σ ) < min{8π, ω n g } δ, modulo the action of the Möbius goup, {Σ } is compact. By esults in [8] ( see also [18]), when W (Σ ) < min{8π, ω n g } δ, the confomal class of Σ must be compact in the moduli space. Moeove, by Li-Yau s inequality [1], Σ is an embedding when W (Σ ) < 8π. When f has no banches at, Theoem 1.1 and Theoem 1.3 can be deduced diectly fom the emovability of singulaity [9] and the classification of Willmoe sphee in S 3 [, 11]. In fact, the esults in [] imply the following: Lemma 1.6. Let f : S R 3 be a Willmoe immesion. If f has no tansvesal selfintesectiones, then f is an embedding and f(s ) is a ound sphee. Then, to get Theoem 1.1 and 1.3, we only need to pove f has no banches at. Fo this sae, we will pove the following: Lemma 1.7. Let f : C \ D R R 3 be a smooth confomal complete embedding with Then A L (C\D R ) < +, f(z) A(z) < +. z + θ ( f(µ g C \ D R ), ) = 1. Lemma 1.8. Let f : C \ D R R 3 be a smooth confomal complete immesion with A L (C\D R ) < +, f(z) A(z) < +. z + If thee exists embedding φ : D R \ D R R 3, which conveges to f 0 in C (D R \ D R ) fo any R > R, then θ ( f(µ g C \ D R ), ) = 1. Acnowledgements. The autho would lie to than Pof. Xiang Ma and Pof. Peng Wang fo stimulating discussions.

3 SOME REMARKS ON WILLMORE SURFACES EMBEDDED IN R 3 3. Complete Willmoe embedding of C in R 3 with A < + In this section, we will pove Lemma 1.7, and use it to pove Theoem The poof of Lemma 1.7. By Theoem 4..1 in [15], we may assume (.1) g = e u g euc, with u = m log z + ω, whee m is a nonnegative intege and z ω(z) exists. Moeove, we have f ew( ) (.) = z + z m+1 m + 1. Also by Theoem 4..1 in [15], we can obtain (.3) θ ( f(µ g C \ D R ), ) = m + 1. Let f (z) = f( z), whee m+1 +. Let H, A be the mean cuvatue and the second fundamental fom of f espectively. By (1.1), Since f = z m e ω( z), we have f = 1 H f. f L (D \D 1 ) C()W (f, D \D 1 ), and + Noting that W (f, D \ D 1 ) W (f) and f (1) eω( ), we get m+1 f L (D \D 1 ) + f W 1, (D \D 1 ) < C(). z m f e ω( ) C 0 (D \D 1 ) Applying elliptic estimates, we have f W, (D \D 1 ) < C(). Thus we may assume f conveges to f 0 wealy in W, (D \ D 1 ). Then we may assume df df conveges to df 0 df 0 in L q (D \ D 1 ) fo any q > 0. Noting that df df = z m e ω( z) g euc, we get df 0 df 0 = z m e ω( ) g euc. Let A 0 be the second fundamental fom of f 0. Obviously, A 0 A = 0, D + \D 1 C\D then C A 0 = 0 and the imagine f 0 is in a plane. Without loss of geneality, we may assume w( ) = 0 and f 0 = (z m+1, c). Next, we pove m = 0 by contadiction. Assume m > 0. By (.), when z D \ D 1, A (z) = m+1 A( z) = m+1 f( z) f( z) A( z) < C(). Then f L (D \D 1 ) < C and f conveges in fact in C 1 (D \ D 1 ). = 0.

4 4 Y. LI Let and If we set f = (ϕ, f 3 ), then ϕ z m+1, f 3 c in C 1 (D \ D 1 ). ( Σ = f (C \ D R ) (D 4 \ D 1 4 ) ) R, F (x 1, x, x 3 ) = (x 1 ) + (x ). Then F is C 1 -smooth on Σ with no citical points when is sufficiently lage. Obviously, {y Σ : F (y) = 1} consists of compact C 1 smooth 1-dimensional manifolds. Since ϕ z m+1 and f is an embedding, {z : F = 1} has at least components. Let {F = 1} = Γ 1 Γ Γ m, whee Γ i ae components of {F = 1} and m. Let φ(, t) be the flow geneated by F / F and put Ω i = φ(γ i, [ 1, 1]). Then Ω i = { F 1 }, and Ω i Ω j =. i That is to say that { F 1 } has at least components, and on each component Ω i, we can find y i such that F (y i ) = 1. Let y i = f (z i ). Recall that fo any fixed small ɛ, when is sufficiently lage, we have We may assume z 1, z D 1+ɛ \ D 1 ɛ ɛ ϕ (z i ) z i m+1 < ɛ, i = 1,. such that ɛ 1, and D 1+ɛ \ D 1 ɛ {z : 3 ϕ (z) 3 4 }. Tae a cuve γ such that γ([0, 1]) D 1+ɛ \ D 1 ɛ, and γ(0) = z 1, γ(1) = z. Then f (γ(0)) = y 1, f (γ(1)) = y, and f (γ) i Ω i. It is a contadiction to the fact that Ω 1 and Ω ae diffeent components... The poof of Theoem 1.1. By a esult of Hube[7], we may assume f to be confomal. Without loss of geneality, we assume f(0) = 0. We may assume A L (C\B R ) < ɛ. Then by Theoem.10 in [9], A L (B \B (0)) < C A L (B 4 \B (0)) wheneve > R. Let Σ be the image of embedding f : C R 3. We deduce fom Lemma 1.7 that µ Σ (B R ) = 1. R + πr Let y 0 / Σ and I(y) = y y 0 y y 0. By Lemma 4.3 in [10], I(Σ) can be extended to a smooth closed suface. It is easy to chec that I(Σ) is an embedded Willmoe sphee. By Lemma 1.6, I(Σ) must be a ound sphee, which implies that Σ is a plane. Then we get Theoem 1.1.

5 SOME REMARKS ON WILLMORE SURFACES EMBEDDED IN R Compactness of a Willmoe embedding sequence in R 3 In this section, we fist pove Lemma 1.8, then pove Theoem 1.4. Since the poof of Theoem 1.3 is vey simila to Theoem 1.1, we omit it The poof of Lemma 1.8. We assume g = e u g euc, with u = m log z + ω, whee m is a nonnegative intege and z ω(z) exists. Simila to the poof of Lemma 1.7, we let f 0,n (z) = f(nz), whee n m+1 n +. we may assume f 0,n conveges to (z m+1, c) in C 1 (D 1 \ D ). Recall that φ conveges to f in C 1. Then, we can find n, such that φ n ( n z) conveges to (z m+1, c). Then using the the aguments simila as we pove Lemma 1.7, we can finish the poof of Lemma The poof of Theoem 1.4. Let f be confomal immesion of (Σ, h ) into R 3, whee h is a smooth metic with constant cuvatue. When the genus of Σ is 1, we assume µ(h ) = 1. Since the confomal stuctue induced by h conveges in the moduli space, we may assume h conveges smoothly to h 0. By esults in [8], we may find Möbius tansfomation σ and a finite set S, such that σ(f ) = 1 and σ (f ) conveges in W, loc (Σ \ S, h 0), whee S = {p Σ : 0 + B h 0 (z) A f 8π}. Let f 0 be the it, which is a banched W, -confomal immesion. Thus f 0 is continuous on Σ. The following theoems will be useful, see [5], [17] fo poofs espectively. Theoem 3.1. Let g, g be smooth Riemannian metics on a suface M, such that g g in C s,α (M), whee s N, α (0, 1). Then fo each p M thee exist neighbohoods U, U and smooth confomal diffeomophisms ϕ : D U, such that ϑ ϑ in C s+1,α (D, M). Theoem 3.. Let f : D R n be a confomal immesion with g f = e u g euc. Assume f is Willmoe. Then thee exists an ɛ 0 > 0 and a λ > 0, such that if A dx < ɛ 0, and u < λ, then whee = ( x 1, x ). Fo simplicity, choose ɛ 0 < 8π. D n L (D ) C(ɛ 0, λ, ) A L (D), Lemma 3.3. Let f : D \ {0} R 3 be a confomal Willmoe immesion with µ f (D) + A L (C\D R ) < +. If thee exist 0 and embedding f : D\D R 3, which conveges to f in C 1 (D\D ) fo any < 1, then fo any sufficiently small θ ( f(µ f D ), 0 ) = 1.

6 6 Y. LI Poof. Set g = e u g euc. Using Poposition 4.1 in [8], f W, (D), and u = m log z + ω(z), whee m is positive intege and ω C 0 (D). Moeove, we have and f(z) f(0) z 0 z m+1 = eω(0) m + 1, θ ( f(µ g D ), 0 ) = m + 1. Without loss of geneality, we assume f(0) = 0. Set f = f, and g = d f d f. Then g = g. Let A 0 = A 1 Hg, which is the taceless f f 4 pat of A. It is well-nown that A 0 dµ g = Ã0 dµ g. D Put g = e ũ g euc. We have ũ = u log f. By Gauss cuvatue equation we get Since D δ \D Ke ũ = = = f f D f D ũ = Ke ũ, ũ D δ + ũ D u D δ + f f D δ f + Ke u f f + D δ \D D δ f f π D f = 0 D u D f f f. D f f f e u(eiθ ) m+1 dθ < C, m f(e iθ ) we get D δ Kdµ g < C(δ). Then D Ã 1 δ dµ f < +. Theefoe, f( ) satisfies the z conditions of Lemma 1.8. Set ˆf(z) = f(1/z), and ĝ = d ˆf d ˆf = e û g euc. We have û(z) = ũ( 1 z ) log z = m log 1 z log f(1 z ) log z. Then (û(z) m log z ) = log f(1 z ) = ω(0) + log(m + 1). z z m+ Applying Lemma 1.8, (.3) and (.1), we get m = 0. 1 z We define We need to pove S is empty. S = {z Σ : 0 + B h 0 (z) A f > ɛ 0 }.

7 SOME REMARKS ON WILLMORE SURFACES EMBEDDED IN R 3 7 Assume S is not empty. Given a point p S, we choose U, U, ϑ, ϑ as in Theoem 3.1, and assume p = 0. We can choose U such that U S = {p}. Let ˆf = f ϑ. Note that ˆf is a confomal map fom D into R 3. Let Put g ˆf = e û g euc, h = e v g euc. Note that 0 is the only point in D which satisfies A 0 ˆf dµ ˆf > ɛ 0. D (z ) + A f = ɛ 0, such that D (z) D (z) Then z 0 and 0. Let f = ˆf ( z+z ) ˆf (z ) λ, whee By Theoem 4.1, λ = diam( ˆf(z + [0, 1/])). u L (D ) C(), (0, 1). Then, by Theoem 3., f conveges smoothly on D 3. 4 Fo any point z 0 D 1, put A f < ɛ 0, z D 1, <. γ (t) = z tz 0, t [0, 1], and τ = diam(f (γ )). Then by Theoem 4.1 and Theoem 3., f f (z 0) τ conveges smoothly. Since f conveges in D 3, we may assume τ τ 0 > 0. Then f conveges smoothly on D 3 (z 0 ). Thus f 4 4 conveges smoothly on D 1. In this way, we can pove that a subsequence of f conveges smoothly on D R fo any R. Let f 0 be the it. Then u 0 L loc (C) and D A f 0 = ɛ 0. Obviously, f 0 is pope. If diam(f 0) = +, then f 0 is noncompact and complete. Then by Theoem 1.3, f 0 is a plane which implies that A D f 0 = 0. A contadiction. So, diam(f 0) < +, then by Simon s inequality [19], µ(f 0) < +. By Poposition 4.1 in [8], f 0 can be consideed as a continuous map fom S into R 3. Now, we set ˆf (z) = ˆf (z + z) and and S( ˆf ) = {z C \ {0} : 0 + D (z) Γ(θ 1, θ, t) = {te iθ : θ 1 θ θ }. Since S( ˆf ) is a finite set, we can choose θ 1 < θ, such that ( ) (3.1) t [,] Γ(θ 1, θ, t) S( ˆf ) =. A ˆf > ɛ 0 },

8 8 Y. LI Tae t [, ], such that By Poposition 4.1 in [8], Then Let and λ = diam( ˆf (Γ(θ 1, θ, t ))) = diam( ˆf t 0 (Γ(θ 1, θ, t))) = 0, + t 0, and t +. inf diam( ˆf (Γ(θ 1, θ, t))). t [,] f = ˆf (t z + z ) ˆf (t e iθ 1 + z ) λ S({f }) = {z C \ {0} : diam(f t (Γ(θ 1, θ, t))) = D (z) A ˆf > ɛ 0 }. By (3.1) and Theoem 3. and Theoem 4.1, f conveges smoothly nea Γ(θ 1, θ, 1). Following the method we get f 0, we obtain that f conveges smoothly on any compact subset of C \ ({S(f )} {0}). Let f 0 be the it. Then (3.) diam(f 0 (Γ(θ 1, θ, t))) = inf diam(f 0 (Γ(θ 1, θ, t))). t (0, ) Then µ f (D 0 (0)) = and µ f (C \ D 0 (0)) = fo any. Othewise, by Poposition 4.1 in [8], diam(f 0 (Γ(θ 1, θ, t))) = 0, o diam(f 0 (Γ(θ 1, θ, t))) = 0. t 0 t + It contadicts (3.). Thus f 0 is complete, noncompact and has at least ends. Now, choose y 0 such that d(y 0, f (Σ) ˆf (t e iθ 1 + z ) ) > δ > 0 λ Set I = y y 0 y y 0. Then I(f ) conveges to I(f 0 ) smoothly on any compact subset of C \ ({0} S({f })). We will follow the agument of Kuwet and Schätzle (see the poof of Theoem 5.1 in [10]) to pove that I(f 0 ) is in fact a smooth immesion of S in R 3. Fo any small and any z S({f }) {0}, since I(f ) is Willmoe on D (z) and conveges smoothly to I(f 0 ) on D (z), we get Res(I(f 0 ), z) = 0. Then, by Lemma 4.1 in [10] (see also Theoem I.6 in [17]) and Lemma 3.3, I(f 0 ) is a smooth Willmoe embedding on D (z). Moeove, fo a lage R, since I(f ) is Willmoe on D R and conveges smoothly on D R, Res(I(f 0 ), ) is also 0. Then I(f 0 )( 1) is a smooth Willmoe embedding on D z 1. R Theefoe, I(f 0 ) can be consideed as a smooth confomal immesion fom S into R 3. Obviously, I(f 0 ) has no tansvesal self-intesections. By Lemma 1.6, I(f 0 ) must be a ound sphee. It contadicts the fact that f 0 has at least ends. Hence we get S =.

9 SOME REMARKS ON WILLMORE SURFACES EMBEDDED IN R 3 9 Then, using the agument in [8], we get u L (Σ) < C (this can also be deduced fom Theoem 4.1). Given a point p Σ, we choose U, U, ϑ, ϑ as in Theoem 3.1, and assume p = 0. Let ˆf = f ϑ, which is confomal. Then we can choose an, such that D A ˆf < ɛ 0. Using Theoem 3., ˆf conveges smoothly on D. We can choose to be sufficiently small, such that thei exists p, such that B h 0 p (p) ϕ (D ). Thus f conveges smoothly on B h 0 p (p). 4. Appendix The poof of the following theoem can be found in [13]. But fo the convenience of the eades, we povide a poof in this appendix. Theoem 4.1. Let f : D R n be a smooth confomal immesion which satisfies 1) A D f dµ f < γ n τ, whee τ > 0 and { 8π when n = 3 γ n = 4π when n 4. ) f (D) can be extended to a closed immesed suface Σ with Σ A f dµ f < Λ. Tae a cuve γ : [0, 1] D, and set λ = diam f (γ). Then we can find a subsequence of f = f f (γ(0)) λ which conveges wealy in W, loc (D). Let df df = e u (dx 1 dx 1 + dx dx ). (4.1) u L (D ) < C(). Poof. Put Σ = Σ f (γ(0)) λ. We have two cases: Case 1: diam(f ) < C. By inequality (1.3) in [19] with ρ =, Σ Bσ(γ(0)) C fo any σ σ > 0. Hence we get µ(f ) < C by taing σ = diam(f ). Then by Helein s convegence theoem [6, 8], f, conveges wealy in Wloc (D). Since diam f (γ) = 1, the wea it is not tivial and (4.1) holds. Case : diam(f ) +. We tae a point y 0 R n and a constant δ > 0, s.t. Let I = y y 0 y y 0, and B δ (y 0 ) Σ =, f = I(f ), Σ = I(Σ ). By confomal invaiance of Willmoe functional [3, 0], we have Σ. A Σ dµ Σ = Σ A Σ dµ Σ < Λ.

10 10 Y. LI Since Σ B 1 (0), also by (1.3) in [19], we get µ(f δ S({f }) := {p D : Then f conveges wealy in W, loc 0 + D ( p) ) < C. Let A f dµ f γ n}. (D \ S(f )). does not convege to a point by assumption. If f Next, we pove that f conveges to a point in W, loc (D \ S(f )), then the it must be 0, fo diam (f ) conveges to +. By the definition of f, we can find a δ 0 > 0, such that f (γ) B δ 0 (0) =. Thus fo any p γ([0, 1]) \ S(f ), f will not convege to 0. A contadiction. Then we only need to pove that f, conveges wealy in Wloc (D, Rn ). Let f 0 be the it of f which is a banched immesion of D in Rn. Let S = f 1 0 ({0}), which is isolate. Note that fo any z 0 S, thee exits m > 0, such that f(z) f(z 0 ) > 0. z z 0 0 z z 0 m Fist, we pove that fo any Ω D \ (S S({f })), f conveges wealy in W, (D, R n ): Since f 0 is continuous on Ω, we may assume dist(0, f 0 (Ω)) > δ > 0. Then dist(0, f (Ω)) > δ when is sufficiently lage. Noting that f = f + y f 0, we get that f conveges wealy in W, (Ω, R n ). Next, we pove that fo each p S S({f }), f also conveges in a neighbohood of p. Let g f = e u geuc. Since f W, conf (D 4δ(p)) with A D 4δ (p) f dµ f < 8π τ when δ is sufficiently small and sufficiently lage, by the aguments in [8], we can find a v solving the equation v = K f e u, z Dδ (p) and v L (D δ (p)) < C. Since f conveges to a confomal immesion in D 4δ \ D 1 δ(p), we may assume that 4 u L (D δ \D δ (p)) < C. Then u v is a hamonic function with u v L ( D δ (p)) < C, then we get u (z) v (z) L (D δ (p)) < C fom the Maximum Pinciple. Thus, u L (D δ (p)) < C, which implies f L (D δ (p)) < C. By the equation f = eu Hf, and the fact that e u Hf L (D δ (p)) < e u L (D δ (p)) H f dµ f, D δ (p) we get f W 1, (D δ (p)) < C. Thus, a subsequence of f conveges wealy in W, (D ) fo any (0, 1). Then, by Helein s convegence theoem, we can get (4.1). Refeences [1] Y. Benad and T. Riviée: Enegy quantization fo Willmoe sufaces and applications. Ann. of Math. () 180 (014), [] R. Byant: A duality theoem fo Willmoe sufaces, J. Diffeential Geom., 0 (1984), [3] B. Y. Chen: Some confomal invaiants of submanifolds and thei applications, Boll. Un. Mat. Ital. 10 (1974),

11 SOME REMARKS ON WILLMORE SURFACES EMBEDDED IN R 3 11 [4] J. Chen and T. Lamm: A Benstein type theoem fo entie Willmoe gaphs. J. Geom. Anal. 3 (013), [5] D. DeTuc and J. Kazdan: Some egulaity theoems in Riemannian geomety. Ann. Sci. cole Nom. Sup. (4) 14 (1981), [6] F. Hélein: Hamonic maps, consevation laws and moving fames. Tanslated fom the 1996 Fench oiginal. With a foewod by James Eells. Second edition. Cambidge Tacts in Mathematics, 150. Cambidge Univesity Pess, Cambidge, 00. [7] A. Hube: On subhamonic functions and diffeential geomety in the lage, Comment. Math. Helv. 3 (1957), [8] E. Kuwet and Y. Li: W, -confomal immesions of a closed Riemann suface into R n. Comm. Anal. Geom. 0 (01), [9] E. Kuwet and R. Schätzle: The Willmoe flow with small initial enegy. J. Diffeential Geom., 57 (001), [10] E. Kuwet and R. Schätzle: Removability of point singulaities of Willmoe sufaces, Ann. of Math. 160 (004), [11] T. Lamm and H. Nguyen: Banched willmoe sphees. J. Reine Angew. Math. 701 (015), [1] P. Li and S.T. Yau: A new confomal invaiant and its applications to the Willmoe conjectue and the fist eigenvalue on compact sufaces, Invent. Math 69 (198), [13] Y. Li: Wea it of an immesed suface sequence with bounded Willmoe functional. axiv: [14] Y. Luo and J. Sun: Remas on a Benstein type theoem fo entie Willmoe gaphs in R 3. J. Geom. Anal. 4 (014), [15] S. Mülle and V. Šveá: On sufaces of finite total cuvatue, J. Diffeential Geom. 4 (1995), [16] J. Péez and A. Ros: Popely embedded minimal sufaces with finite total cuvatue. The global theoy of minimal sufaces in flat spaces (Matina Fanca, 1999), 15-66, Lectue Notes in Math., 1775, Spinge, Belin, 00. [17] T. Riviée: Analysis aspects of Willmoe sufaces. Invent. Math. 174 (008), no. 1, [18] T. Riviée: Lipschitz confomal immesions fom degeneating Riemann sufaces with L -bounded second fundamental foms. Adv. Calc. Va. 6 (013), [19] L. Simon: Existence of sufaces minimizing the Willmoe functional, Comm. Anal. Geom. 1 (1993), [0] T. J. Willmoe: Total Cuvatue in Riemannian Geomety, John Wiley & Sons, New Yo (198).

RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION

RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION JINGYI CHEN AND YUXIANG LI Abstact. We show that a smooth adially symmetic solution u to the gaphic Willmoe suface equation is eithe