Skip to main content

\(C^{1,\alpha }\)-regularity for surfaces with \(H\in L^p\)


In this paper, we prove several results on the geometry of surfaces immersed in \(\mathbb {R}^3\) with small or bounded \(L^2\) norm of \(|A|\). For instance, we prove that if the \(L^2\) norm of \(|A|\) and the \(L^p\) norm of \(H\), \(p>2\), are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded \(L^2\) norm of \(|A|\), not necessarily small, then such a disk is graphical away from its boundary, provided that the \(L^p\) norm of \(H\) is sufficiently small, \(p>2\). These results are related to previous work of Schoen–Simon (Surfaces with quasiconformal Gauss map. Princeton University Press, Princeton, vol 103, pp 127–146, 1983) and Colding–Minicozzi (Ann Math 160:69–92, 2004).

This is a preview of subscription content, access via your institution.


  1. Allard, W.K.: On the first variation of a varifold. Ann. Math. 95, 417–491 (1972). (MR0307015, Zbl 0252.49028)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bourni, T., Tinaglia, G.: Curvature estimates for surfaces with bounded mean curvature. Trans. Am. Math. Soc. 364(11), 5813–5828 (2012). (MR2946933)

    Article  MATH  MathSciNet  Google Scholar 

  3. Choi, H.I., Schoen, R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. Invent. Math. 81, 387–394 (1985). (MR0807063, Zbl 0577. 53044)

    Article  MATH  MathSciNet  Google Scholar 

  4. Colding, T.H., Minicozzi II, W.P.: The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks. Ann. Math. 160, 69–92 (2004). (MR2119718, Zbl 1070.53032)

    Article  MATH  MathSciNet  Google Scholar 

  5. Colding, T.H., Minicozzi II, W.P.: A course in minimal surfaces. Graduate studies in mathematics. American Mathematical Society, Providence, RI (2011)

    MATH  Google Scholar 

  6. Fiala, F.: Le problème des isopérimètres sur les surfaces ouvertes à courbure positive. Comment. Math. Helv. 13, 293–346 (1941)

    Article  MathSciNet  Google Scholar 

  7. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd edn. Springer, New York (1983). (MR0737190, Zbl0562.35001)

    Book  MATH  Google Scholar 

  8. Hartman, P.: Geodesic parallel coordinates in the large. Am. J. Math. 86, 705–727 (1964). (MR0173222)

    Article  Google Scholar 

  9. Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. XXVII, 715–727 (1974)

    MathSciNet  Google Scholar 

  10. Rosenberg, H.: Some recent developments in the theory of minimal surfaces in 3-manifolds. In: 24th Brazilian Mathematics Colloquium, Instituto de Matematica Pura e Aplicada (IMPA). IMPA Mathematical Publications, Rio de Janeiro (2003) (MR2028922 (2005b:53015), Zbl 1064.53007)

  11. Schoen, R., Simon, L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34(6), 741–797 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  12. Schoen, R., Simon, L.: Surfaces with quasiconformal Gauss map. In: Bombieri, E. (ed.) Seminar on minimal submanifolds, Annals of Mathematics Studies, vol. 103, pp. 127–146. Princeton University Press, Princeton (1983)

  13. Shiohama, K., Tanaka, M.: An isoperimetric problem for infinitely connected complete open surfaces. In: Geometry of manifolds (Matsumoto, 1988) Perspect. Math, vol. 8, pp. 317–343. Academic Press, Boston (1989)

  14. Shiohama, K., Tanaka, M.: The length function of geodesic parallel circles. Progress in differential geometry. Adv. Stud. Pure Math. 22, 299–308 (1993). (MR1274955 (95b:53054), Zbl 0799.53052)

    MathSciNet  Google Scholar 

  15. Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Center for Mathematical Analysis, vol. 3. Australian National University, Canberra (1983). (MR0756417, Zbl 546.49019)

  16. Toro, T.: Surfaces with generalized second fundamental form in \(L^2\) are Lipschitz manifolds. J. Differ. Geom. 39(1), 65–101 (1994)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Giuseppe Tinaglia.

Additional information

G. Tinaglia is partially supported by EPSRC Grant No. EP/L003163/1.



For the sake of completeness, in this appendix, we prove two results in differential geometry that are used throughout the paper. In Remark 5.3, we also discuss the \(C^{1,\alpha }\) regularity.


$$\begin{aligned} \Omega :=\{(x_1,x_2)\in \mathbb {R}^2| a^2<x_1^2+x_2^2<b^2\} \end{aligned}$$

for certain \(b>a>0\) and let \(\mathcal A\) denote the graph above \(\Omega \) of a smooth function \(u\). That is \(u\in C^\infty (\Omega )\) and \({{\mathrm{graph}}}u=\mathcal A\).

Lemma 5.1

Assume that

$$\begin{aligned} |Du|\le r\le 1 \text { and} \int _\mathcal A|A|^2\,\mathrm{d}\mathcal {H}^2\le \varepsilon \end{aligned}$$

Then, there exists \(\rho \in (a,b)\) for which

$$\begin{aligned} \int _{u(S_\rho )}k \,\mathrm{d}s\le 2\pi \left( 1+r\sqrt{2}+ \left( \frac{2\varepsilon b}{b-a}\right) ^\frac{1}{2}\right) , \end{aligned}$$

where \(k\) is the curvature of the curve \({{\mathrm{graph}}}u|_{S_\rho }\) and \(S_\rho =\{(x_1, x_2): x_1^2+x_2^2=\rho ^2\}\).


Recall that in graphical coordinates

$$\begin{aligned} A_{ij}(x, u(x))=\left( \frac{\partial ^2}{\partial x_i \partial x_j}\left( x_1, x_2, u(x_1, x_2)\right) \right) ^\bot =\left( 0,0, D_{ij} u\right) \cdot \nu = \frac{D_{ij} u}{\sqrt{1+|Du|^2}}, \end{aligned}$$

where \(A_{ij}\), \(i=1,2\), are the coefficients of the second fundamental form, and also that

$$\begin{aligned} |A|^2=A_{ij}A_{kl}g^{ik}g^{il}, \end{aligned}$$

where \(g\) is the induced metric. We have that

$$\begin{aligned} |D^2 u|^2=\sum _{i,j=1}^2|D_{ij} u|^2 \le |A|^2 (1+|Du|^2)^3 \end{aligned}$$

(see for example [5]). On \(\Omega \), we are assuming that

$$\begin{aligned} |Du(x)|\le r \,, \quad \forall x\in \Omega \end{aligned}$$

and this, together with the area formula, gives

$$\begin{aligned} \int _\Omega |D^2u(x)|^2\,\mathrm{d}x&= \int _\Omega \frac{|D^2u|^2}{(1+|Du|^2)^3} (1+|Du|^2)^3 \mathrm{d}x\\&\le (1+r^2)^\frac{5}{2} \int _\Omega \frac{|D^2u|^2}{(1+|Du|^2)^3} \sqrt{1+|Du|^2} \mathrm{d}x\\&= (1+r^2)^\frac{5}{2}\int _{\mathcal A}|A|^2\,\mathrm{d}\mathcal {H}^2\le 2(1+r)\varepsilon . \end{aligned}$$

By the coarea formula, we can pick \(\rho \in (a, b)\), so that

$$\begin{aligned} \int _{S_\rho }|D^2u|^2\,\mathrm{d}x\le \frac{2(1+r)\varepsilon }{b-a}. \end{aligned}$$

Let \(\Gamma = {{\mathrm{graph}}}u|_{S_\rho }\). \(\Gamma \) is a closed curve in \(\mathcal A\) and we want to compute

$$\begin{aligned} \int _\Gamma k\,\mathrm{d}s. \end{aligned}$$

Let \(\gamma :[0,1]\rightarrow S_\rho \) be the following parametrization of \(S_\rho \):

$$\begin{aligned} \gamma (t)=(\rho \cos 2\pi t, \rho \sin 2\pi t) \end{aligned}$$

and consider the parametrization of \(\Gamma \) given by

$$\begin{aligned} f(t)=(\gamma (t), u(\gamma (t)))\,,\,\,t\in [0,1]. \end{aligned}$$

Recall that

$$\begin{aligned} \int _\Gamma k\,\mathrm{d}s=\int _0^1k(f(t)) |f'(t)|\,\mathrm{d}t\le \int _0^1\frac{|f''|}{|f'|}\mathrm{d}t, \end{aligned}$$


$$\begin{aligned} k=\frac{|f'\times f''|}{|f'|^3}\implies k\le \frac{|f''|}{|f'|^2}. \end{aligned}$$


$$\begin{aligned} |f'|^2&= |\gamma '(t)|^2+\left| \frac{\text{ d }}{{\text{ d }}t}u(\gamma (t))\right| ^2\implies \\ (2\pi \rho )^2&= |\gamma '(t)|^2\le |\gamma '(t)|^2+|Du|^2|\gamma '(t)|^2= |f'|^2 \end{aligned}$$


$$\begin{aligned} |f''|&= \left| \left( \gamma ''(t),\frac{\mathrm{d}^2}{\mathrm{d}t^2}u(\gamma (t))\right) \right| \le \left| \gamma ''(t)\right| +\left| \frac{\mathrm{d}^2}{\mathrm{d}t^2}u(\gamma (t))\right| \\&\le \rho (2\pi )^2+\sqrt{2}|D^2u||\gamma '(t)|^2+\sqrt{2}|Du||\gamma ''(t)|\\&\le \rho (2\pi )^2+\sqrt{2}|D^2u|(2\pi \rho )^2+\sqrt{2}r \rho (2\pi )^2, \end{aligned}$$

where we have used the computation:

$$\begin{aligned} \left| \frac{{\text{ d }}^2}{{\text{ d }}t^2}u(\gamma )\right|&= \left| \frac{\text{ d }}{{\text{ d }}t}(Du\cdot \gamma ')\right| \le |\gamma '|^2\left( \sum _{i,j=1}^2 |D_{ij} u|\right) +|Du|\left( |\gamma _1''|+|\gamma _2''|\right) \\&\le |\gamma '|^2\sqrt{2}\left( \sum _{i,j=1}^2 |D_{ij} u|^2\right) ^\frac{1}{2}+|Du|\sqrt{2}|\gamma ''|\\&= |\gamma '|^2\sqrt{2}|D^2u|+|Du|\sqrt{2}|\gamma ''|. \end{aligned}$$


$$\begin{aligned} \int _\Gamma k\,\mathrm{d}s&\le \int _0^1\frac{\rho (2\pi )^2+\sqrt{2}|D^2u|(2\pi \rho )^2+ r\rho \sqrt{2} (2\pi )^2}{2\pi \rho }\mathrm{d}t\\&= 2\pi (1+\sqrt{2}r)+2\sqrt{2}\pi \rho \int _0^1|D^2u(\gamma (t))|\mathrm{d}t. \end{aligned}$$

Using again the area formula, we have

$$\begin{aligned} \int _0^1 |D^2u(\gamma (t))|\mathrm{d}t&= \int _0^1 \frac{|D^2u(\gamma (t))|}{|\gamma '(t)|}|\gamma '(t)|\mathrm{d}t=\frac{1}{2\pi \rho } \int _0^1 |D^2u(\gamma (t))||\gamma '(t)|\mathrm{d}t\\&= \frac{1}{2\pi \rho }\int _{S_\rho }|D^2u(x)|\mathrm{d}x\le \frac{1}{2\pi \rho }\left( \int _{S_\rho }|D^2u(x)|^2\mathrm{d}x\right) ^\frac{1}{2}|S_\rho |^\frac{1}{2}\\&\le \frac{1}{2\pi \rho } \left( \frac{4\pi \rho (1+r)\varepsilon }{ b-a}\right) ^\frac{1}{2}= \left( \frac{(1+r)\varepsilon }{\pi \rho ( b-a)}\right) ^\frac{1}{2}. \end{aligned}$$


$$\begin{aligned} \int _\Gamma k\, \mathrm{d}s&\le 2\pi (1+r\sqrt{2})+2\sqrt{2}\pi \rho \left( \frac{(1+r)\varepsilon }{\pi \rho ( b-a)}\right) ^\frac{1}{2}=2\pi \left( 1+r\sqrt{2}+\left( \frac{2(1+r)\varepsilon \rho }{\pi (b-a)}\right) ^\frac{1}{2}\right) \\&\le 2\pi \left( 1+r\sqrt{2}+\left( \frac{2\varepsilon b}{b-a}\right) ^\frac{1}{2}\right) . \end{aligned}$$

\(\square \)

Lemma 5.2

Let \(\mathcal {B}_R:=\mathcal {B}_R(x_0) \subset M\setminus \partial M\) and assume that

$$\begin{aligned} g(x)=\frac{1}{\sqrt{2}}|\nu (x)- e_3|\le r\,,\,\forall x\in \mathcal {B}_R, \end{aligned}$$

for some \(r\in \left[ 0,\frac{1}{\sqrt{2}}\right] \). Then, \(\mathcal {B}_R\) is locally graphical over the plane \(\{x_3=0\}\) with gradient bounded by \(3r\). Moreover, \(\mathcal {B}_R\) contains a graph of a function \(u\) over the disk in the plane \(\{x_3=0\}\) centered at \(\Pi (x_0)\) and of radius \(\rho =\frac{R}{\sqrt{1+(3r)^2}}\); where \(\Pi \) denotes the projection on the plane \(\{x_3=0\}\).


Since \(g(x)\le 1\) for all \(x\in \mathcal {B}_R\), we have that \(\mathcal {B}_R\) is locally a graph over the plane \(\{x_3=0\}\) and at each point \(x=(x_1,x_2, u(x_1, x_2))\in \mathcal {B}_R\), we have

$$\begin{aligned} \nu (x)=\left( -\frac{D_1u}{\sqrt{1+|Du|^2}}, -\frac{D_2u}{\sqrt{1+|Du|^2}},\frac{1}{\sqrt{1+|Du|^2}}\right) \end{aligned}$$

where \(\nu \) is the upward pointing unit normal. We estimate now \(|Du|^2=|D_1u|^2+|D_2u|^2\) using the estimate for \(g\) as follows: Note first that

$$\begin{aligned} g(x)&= \frac{1}{\sqrt{2}}|\nu (x)-e_3|=\frac{1}{\sqrt{2}}\left( \frac{|D_1u|^2}{1+|Du|^2}+\frac{|D_2 u|^2}{1+|Du|^2}+\left( 1-\frac{1}{\sqrt{1+|Du|^2}}\right) ^2\right) ^\frac{1}{2}\\&= \frac{1}{\sqrt{2}}\left( \frac{|Du|^2}{1+|Du|^2}+\frac{1}{1+|Du|^2}+1-2\frac{1}{\sqrt{1+|Du|^2}}\right) ^\frac{1}{2}\\&= \frac{1}{\sqrt{2}}\left( 2-2\frac{1}{\sqrt{1+|Du|^2}}\right) ^\frac{1}{2}=\left( 1-\frac{1}{\sqrt{1+|Du|^2}}\right) ^\frac{1}{2}. \end{aligned}$$

Hence, since \(g(x)\le r\), we get

$$\begin{aligned} g(x)=\left( 1-\frac{1}{\sqrt{1+|Du|^2}}\right) ^\frac{1}{2}&\le r\implies 1-\frac{1}{\sqrt{1+|Du|^2}}\le r^2\implies \\ \sqrt{1+|Du|^2}&\le \frac{1}{1-r^2}\le 1+2 r^2, \end{aligned}$$

with the last inequality being true since \(r\le \frac{1}{\sqrt{2}}\implies r^2-2 r^4\ge 0\). Squaring both sides, we obtain

$$\begin{aligned} 1+|Du|^2\le 1+4r^4+4 r^2\implies |Du|^2\le 9 r^2\implies |Du|\le 3 r. \end{aligned}$$

This finishes the proof of the first part of the lemma.

By the previous discussion, \(\mathcal {B}_R\) is a graph of a function \(u\) around the point \(x_0\). Let \(\rho \) be such that \(u\) is defined on the disk centered at \(\Pi (x_0)\) of radius \(\rho \) in the plane \(\{x_3=0\}\), \(D_\rho (\Pi (x_0))\). Without loss of generality, let \(x_0=0\). We will prove a lower estimate for the radius \(\rho \) of the disk where the function \(u\) is defined. To do this, let \(\rho \) be the maximum such radius. Then, there exists a point \((x_1,x_2)\in \partial D_\rho (0)\), for which \((x_1, x_2, u(x_1, x_2))\in \partial \mathcal {B}_R\), else \(u\) maps \(\partial D_\rho (0)\) in the interior of \(\mathcal {B}_R\) and since \(\mathcal {B}_R\) is locally a graph over the plane \(\{x_3=0\}\) we could increase \(\rho \). Let \(\gamma (t)\) be the path in \(\mathcal {B}_R\) defined by

$$\begin{aligned} \gamma :[0,1]\rightarrow \mathcal {B}_R, \quad \gamma (t)=(tx_1,tx_2,u(t(x_1,x_2))). \end{aligned}$$

The path \(\gamma \) joins \(0\), that is the center of \(\mathcal {B}_R\), with \(x\in \partial \mathcal {B}_R\), therefore it must have length at least \(R\), from which we get

$$\begin{aligned} R\le {{\mathrm{Length}}}(\gamma )=\int _0^1|\dot{\gamma }|\mathrm{d}t\le \int _0^1 \rho \sqrt{1+|Du|^2}\mathrm{d}t \le \rho \sqrt{1+(3r)^2} \end{aligned}$$

which implies that

$$\begin{aligned} \rho \ge \frac{R}{\sqrt{1+(3r)^2}} \end{aligned}$$

and this finishes the proof of the lemma. \(\square \)

Remark 5.3

Standard PDE theory implies that under the hypotheses of Lemma 5.2 and if in addition \(r\le \frac{1}{\sqrt{3}}\) and \(H\in L^p(\mathcal {B}_R)\), \(p>2\), we obtain \(C^{1,\alpha }\) estimates in \(\mathcal {B}_{\frac{R}{2}}\); namely, there exist constants \(\alpha \in (0,1)\) and \(C\), depending on \(r, R^{1-\frac{2}{p}}\int _{\mathcal {B}_R}|H|^p \mathrm{d}\mathcal {H}^2\), and \(p\), such that

$$\begin{aligned} \frac{|\nu (x)-\nu (y)|}{|x-y|^\alpha }\le R^{-\alpha }C,\quad \forall x,y\in \mathcal {B}_{R/2} \end{aligned}$$

To see why the above remark is true, we can assume without loss of generality that \(x_0=0\). Note that by Lemma 5.2, \(\mathcal {B}_R\) contains a graph of a function \(u\) over \(\Omega :=\{(x_1, x_2): x_1^2+x_2^2\le \rho ^2\}\) with \(|Du|\le 3r\) and with

$$\begin{aligned} \rho =\frac{R}{\sqrt{1+ (3r)^2}}\ge \frac{R}{\sqrt{10}}\ge \frac{R}{2}. \end{aligned}$$

Thus, \(\mathcal {B}_{\frac{R}{2}}\subset {{\mathrm{graph}}}u\). Furthermore, \(u\) satisfies the equation

$$\begin{aligned} \sum _{i=1}^2 D_i\left( \frac{D_iu(x)}{\sqrt{1+|Du(x)|^2}}\right) = H(x, u(x)). \end{aligned}$$

By differentiating the above equation, we obtain that \(w= D_ku\), for \(k=1,2\), is a solution to the equation

$$\begin{aligned} \sum _{i,j=1}^2D_i(a^{ij} D_jw)= D_k H \end{aligned}$$

(cf. [7, pages 319–320]) with

$$\begin{aligned} a^{ij}=\frac{\delta _{ij}}{\sqrt{1+|Du|^2}}- \frac{D_iu D_ju}{\sqrt{1+|Du|^2}}. \end{aligned}$$

Note that since \(|Du|\) is bounded, we have that \(H\in L^p(\mathcal {B}_R)\implies H\in L^p (\Omega )\). We can then apply Theorem 8.22 in [7] to obtain that

$$\begin{aligned} {\mathop {\mathrm{sup}}_{x, y\in \Omega }}\frac{|Du(x)- Du(y)|}{|x-y|^\alpha }\le \rho ^{-\alpha } C \end{aligned}$$

for some \(\alpha \in (0,1)\) and with \(C\) and \(\alpha \) depending on \(\mathrm{sup}_{\Omega }|Du|\), \(\rho ^{1-\frac{2}{p}}\Vert H\Vert _{L^p(\Omega )}\) and \(p\), namely on \(r\), \(R^{1-\frac{2}{p}}\Vert H\Vert _{L^p(\mathcal {B}_R)}\) and \(p\), and \(\rho \ge \frac{R}{2}\). Using the formula for \(\nu \) as in (30), we get the required estimate.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bourni, T., Tinaglia, G. \(C^{1,\alpha }\)-regularity for surfaces with \(H\in L^p\) . Ann Glob Anal Geom 46, 159–186 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Embedded Disk
  • Geodesic Ball
  • Gauss Bonnet Theorem
  • Mean Curvature
  • Versus Extrinsic