It was Severi who stated it as a theorem in [Se], whose proof was not correct unfortunately. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. TheDirichlet problem forthe minimal surface problem istofindafunction u of minimal area A(u), as defined in (6) – (7), in the class BV(Ω) with prescribeddataφon∂Ω. These are straightforward generalizations of Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces, respectively. 68 0 obj Mech.14, 1049–1056 (1965), Spruck, J.: Remarks on the stability of minimal submanifolds ofR Barbosa, J. L. (et al.) Rend. In fact, all the known proofs are related to some sort of stability: geometric invariant theory in [CH] (in Let M be a minimal surface in the simply-connected space form of constant curvature a, and let D be a simply-connected compact domain with piecewise smooth boundary on M. Let A denote the second fundamental form of M . References His key idea was to apply the stability inequality[See §1.2] to different well chosen functions. The minimal surface equation 4/3 Calibrations 4/5 First variation and flux 4/8 Monotonicity 4/10 Extended Monotonicity 4/12 Bernstein's theorem 4/15 Stability 4/17 Stability continued 4/19 Stability stability stability 4/22 Bernstein theorem version 2 4/24 Weierstrass representation 4/26: Twistors 4/29 the second variation of the area functional is non-negative. Amer. Theorem 1.5 (Severi inequality). Math. Guisti [3] found nonlinear entire minimal graphs in Rn+1. minimal surfaces: Corollary 2. Indeed, the role of … Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. For the systems that concern us in subsequent chapters, this area property is irrelevant. The Gauss-Bonnet Theorem (see Singer and Thorpe’s book). 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. For an immersed minimal surface in R3, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. Proof. << It is well-known that a minimal graph of codimension one is stable, i.e. Barbosa, João Lucas (et al.) Math. Anal.58, 285–307 (1975), Peetre, J.: A generalization of Courant's nodal domain theorem. of Math.88, 62–105 (1968), Schiffman, M.: The Plateau problem for non-relative minima. Then!2 X=k 4˜(O X): Let us brie y introduce the history of this inequality. 43o �����lʮ��OU�-@6�]U�hj[������2�M�uW�Ũ� ^�t��n�Au���|���x�#*P�,i����˘����. Circ. Publication: Abstract and Applied Analysis. Learn more about Institutional subscriptions, Barbosa, J.L., do Carmo, M.: On the size of a stable minimal surface inR Anal.52, 319–329 (1973), Morrey, Jr., C.B., Nirenberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. In recent years, a lot of attention has been given to the stability of the isoperimetric/Wul inequality. Arch. Stability of Minimal Surfaces and Eigenvalues of the Laplacian. Moreover, the minimal model is smooth. The Wul inequality states that, for any set of nite perimeter EˆRn, one has F(E) njKj1n jEj n 1 n; (1.1) see e.g. [SSY], [CS] and [SS]. Jaigyoung Choe's main interest is in differential geometry. In particular, we consider the space of so-called stable minimal surfaces. Processing of Telemetry Data Generated By Sensors Moving in a Varying Field (M113) D.J. Since minimal graphs are area-minimizing , it is natural to consider stable mini-mal hypersurfaces in Rn+1. J. Math.98, 515–528 (1976), Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. Z.162, 245–261 (1978), Barbosa, J.L., do Carmo, M.: A necessary condition for a metric inR https://doi.org/10.1007/BF01215521, Over 10 million scientific documents at your fingertips, Not logged in A theorem of Micallef, which makes use of the complex stability inequality, states that any complete parabolic two-dimensional surface in four-dimensional Euclidean space is holomorphic Of course the minimal surface will not be stationary for arbitrary changes in the metric. Z. A Reverse Isoperimetric Inequality and Extremal Theorems 3 1. A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ (K S) (). n+1 to be isometrically and minimally immersed inM a time symmetric Cauchy surface, then θ+ = 0 if and only if Σ is minimal. It became again as a conjecture in [Ca,Re]. Remarks. We do not know the smallest value of a for which A-aK has a positive solution. Stable minimal surfaces have many important properties. So we get the minimal surface equation (MSE): div(ru p 1 + jruj2) We call the solution to this equation is minimal surface. Pogorelov [22]). $\begingroup$ The problem asks for the stability of the minimal surface. 2 [18] uses this notation for the intersection number mod 2 14 Proof. This notion of stability leads to an area inequality and a local splitting theorem for free boundary stable MOTS. The UConn Summer School in Minimal Surfaces, Flows, and Relativity is a focused one-week program for graduate students and recent PhDs in geometric analysis, from 16th to 20th, July 2018. The Zero-Moment Point (ZMP) [1] criterion, namely that We link these stability properties with the surface gravity of the horizon and/or to the existence of minimal sections. 3 The Stability Estimate In this section we prove an estimate on the integral of the curvature which will be used in the proof of Bernstein’s theorem. In [10] do Carmo and Peng gave uis minimal. Destination page number Search scope Search Text Search scope Search Text The key underlying property of the local versions of the inequality is the notion of stability, both for minimal hypersurfaces and for … Using the inequality of the Lemma for m = 2, we can improve the stability theorem of Barbosa and do Carmo [2]. ... 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. To learn the Moser iteration technique, follow [GT]. 2 In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold M n in the hyperbolic space H n + m in order to show that M n is totally geodesic. We will usually assume that our curves c: (a;b)! A stability criterion can be seen as a set of inequality constraints describing the conditions under which these equalities are preserved. Tax calculation will be finalised during checkout. Pure Appl. Gauss curvature for stable minimal surfaces in R3, which yielded the Bern-stein theorem for complete stable minimal surfaces in R3. Theorem 3.1 ([27, Theorem 0.2]). Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. Math. The case involving both charge and angular momentum has been proved recently in [25]. Curves with weakly bounded curvature Let § be 2-manifold of class C2. the inequality jSj 4pQ2 was proved for suitable surfaces. Ci. If (M;g) has positive Ricci curvature, then cannot be stable. Barbosa and do Carmo showed [9] that an orientable minimal surface in R3 for which the area of the im-age of the Gauss map is less than 2 π is stable. For basics of hypersurface geometry and the derivation of the stability inequality, Simons’ identity and the Sobolev inequality on minimal hypersurfaces, [S] is an excellent reference. Pogorelov [22]). Amer. In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the space BV ... establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. minimal surface M is a plane (Corollary 4). A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ(K S ) ( [16]). The stability inequality (where D is the covariant derivative with respect to the Riemannian metric h) ⑤Dα⑤ 2 … We note that a noncompact minimal surface is said to be stable if its index is zero. The Sobolev inequality (see Chapter 3). Math. Assume that is stable. Z.144, 169–174 (1975), Departamento de Matematica, Universidade Federal do Ceará, Fortaleza Ceará, Brasil, Instituto de Matematica Pura e Aplicada, Rua Luiz de Camões 68, 20060, Rio de Janeiro, R.J., Brasil, You can also search for this author in Preprint, Chern, S.S.: Minimal submanifolds in a Riemannian manifold. These are minimal surfaces which, loosely speaking, are area-minimizing. Anal.45, 194–221 (1972), Lawson, Jr., B.: Lectures on Minimal Submanifolds. Let X be a smooth minimal surface of general type over kand of maximal Albanese dimension. A theorem of Micallef, which makes use of the complex stability inequality, states that any complete parabolic two-dimensional surface in four-dimensional Euclidean space is holomorphic Mat. Speaker: Chao Xia (Xiamen University) Title: Stability on … A minimal surface is called stable if (and only if) the second variation of the area functional is nonnegative for all compactly supported deformations. Classify minimal surfaces in R3 whose Gauss map is one to one (see Theorem 9:4 in Osserman’s book). On the other hand, we can use either the Gauss–Bonnet theorem or the Jacobi equation to get the opposite bound. volume 173, pages13–28(1980)Cite this article. Stability inequality : Carlo Schmid [CM11] Ch.1 §4 and Ch.1 §5 until Lemma 1.19 included : S.02.a: Bernstein problem : Daniel Paunovic [CM11] conclusion of Ch.1 §5 : S.02.b: 28.10. Exercise 6. On the basis of this inequality, we obtain sufficient conditions for the existence and non-existence of a MOTS (along with outer trapped surfaces) in the domain, and for the existence of a minimal surface in its Jang graph, expressed in terms of various quasi-local mass quantities and the boundary geometry of the domain. n. Math. Pages 441-456. interval (δ, 1], where δ > 0, and the extrinsic curvature of the surface satisfies the inequality \K e \ > 3(1 - δ)2/2δ. of Math.40, 834–854 (1939), Smale, S.: On the Morse index theorem. On the Size of a Stable Minimal Surface in R 3 Pages 115-128. 98, 515–528 (1976) Google Scholar. It is well known that do Carmo and Peng [10], in 1979, proved that a complete stable minimal surface in R3 must be a plane (cf. It is well known that do Carmo and Peng [10], in 1979, proved that a complete stable minimal surface in R3 must be a plane (cf. We also obtain sharp upper bound estimates for the first eigenvalue of the super stability operator in the case of M is a surface in H 4. Index, vision number and stability of complete minimal surfaces. In chemical reactions involving a solid material, the surface area to volume ratio is an important factor for the reactivity, that is, the rate at which the chemical reaction will proceed. Ann. Nonlinear Sampled-Data Systems and Multidimensional Z-Transform (M112) A. Rault and E.I. A Stability Inequality For a Class of Nonlinear Feedback Systems (M114) A.G. Dewey. Rational Mech. [17, 15]. Stability inequality : Carlo Schmid [CM11] Ch.1 §4 and Ch.1 §5 until Lemma 1.19 included : S.02.a: Bernstein problem : Daniel Paunovic [CM11] conclusion of Ch.1 §5 : S.02.b: 28.10. A complex version of the stability inequality for minimal surfaces was derived, in-cluding curvature terms for the case of an underlying space which is not at. 2. Classify minimal surfaces in R3 whose Gauss map is … Destination page number Search scope Search Text Search scope Search Text Then A 4πQ2 , (43) where A is the area of S and Q is its charge. In §5 we prove a theorem on the stability of a minimal surface in R4, which does not have an analogue for 3-dimensional spaces. The inequality was used by Simon in [Si] to show, among other things, that stable minimal hypercones of R n + 1 must be planar for n ≤ 6 and it was subsequently used to infer curvature estimates for stable minimal hypersurfaces, generalizing the classical work of Heinz [He], cf. 162, … If f: U R2!R is a solution of the minimal surface equation, then for all nonnegative Lipschitz functions : R3!R with support contained in U R, Z graph(f) jAj2 2d˙ C Z graph(f) jr graph(f) j 2d˙
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