https://doi.org/10.1007/978-3-642-25588-5_15. I.M.P.A., Rio de Janeiro: Instituto de Matematica Pura e Applicada 1973, Lichtenstein, L.: Beiträge zur Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung von elliptischem typus. A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ(K S ) ( [16]). For the systems that concern us in subsequent chapters, this area property is irrelevant. 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. Theorem 1.5 (Severi inequality). (i) The maximal quotients of the helicoid and the Scherk's surfaces … Math.10, 271–290 (1957), Osserman, R., Schiffer, M.: Doubly-connected minimal surfaces. 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. J. Analyse Math.19, 15–34 (1967), Kaul, H.: Isoperimetrische Ungleichung und Gauss-Bonnet-Formel fürH-Flächen in Riemannschen Mannigfaltigkeiten. Barbosa, J.L., do Carmo, M.: A proof of the general isoperimetric inequality for surfaces. Circ. ... 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. Recall that if X is a minimal surface of general type over k, and ω X is the canonical bundle of X, then the Noether inequality asserts that h 0 (ω X) ⩽ 1 2 … the inequality jSj 4pQ2 was proved for suitable surfaces. Part of Springer Nature. The slope inequality asserts that ω2 f ≥ 4g −4 g deg(f∗ωf) for a relative minimal fibration of genus g ≥ 2. The key underlying property of the local versions of the inequality is the notion of stability, both for minimal hypersurfaces and for … References Indeed, the role of … 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. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. 98, 515–528 (1976) Google Scholar. Helv.46, 182–213 (1971), Bol, G.: Isoperimetrische Ungleichungen für Bereiche und Flächen. of Math.40, 834–854 (1939), Smale, S.: On the Morse index theorem. These are minimal surfaces which, loosely speaking, are area-minimizing. ;�0,3�r˅+���,cJ�"MbF��b����B;�N�*����? 3 %���� : Complete minimal surfaces with total curvature −2π. 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. minimal surface in hyperbolic space satisfy the following relation (Gauss’ lemma): K = −1− |B|2 2. In the context of multi-contact planning, it was advocated as a generalization ... do not mention how to compute the inequality constraints applying to these new variables. Using the inequality of the Lemma for m = 2, we can improve the stability theorem of Barbosa and do Carmo [2]. Minimal surfaces and harmonic functions : Fabian Jin [Oss86] §4 until Lemma 4.2 included : S.01.b: 15.10. the second variation of the area functional is non-negative. minimal surface M is a plane (Corollary 4). Jury. For the integral estimates on jAj, follow the paper [SSY]. • When S is a K3 surface, Bayer … These are straightforward generalizations of Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces, respectively. On the other hand, we can use either the Gauss–Bonnet theorem or the Jacobi equation to get the opposite bound. Pages 441-456. Jaigyoung Choe's main interest is in differential geometry. 43o �����lʮ��OU�-@6�]U�hj[������2�M�uW�Ũ� ^�t��n�Au���|���x�#*P�,i����˘����. Amer. 3 Stable minimal surfaces and the first eigenvalue The purpose of this section is to obtain upper bounds for the first eigenvalue of stable minimal surfaces, which are defined as follows. Nonlinear Sampled-Data Systems and Multidimensional Z-Transform (M112) A. Rault and E.I. Jber. Z.162, 245–261 (1978), Barbosa, J.L., do Carmo, M.: A necessary condition for a metric inR To learn the Moser iteration technique, follow [GT]. Nashed, M.Zuhair; Scherzer, Otmar. ... A theorem of Hopf and the Cauchy-Riemann inequality. 1See [CM1] [CM2] for further reference. Since minimal graphs are area-minimizing , it is natural to consider stable mini-mal hypersurfaces in Rn+1. 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. Destination page number Search scope Search Text Search scope Search Text Assume that is stable. Z. ... J. Choe, The isoperimetric inequality for a minimal surface with radially connected boundary, MSRI preprint. Pogorelov [22]). uis minimal. Pages 167-182. It is well-known that a minimal graph of codimension one is stable, i.e. /Filter /FlateDecode A strong stability condition on minimal submanifolds Chung-Jun Tsai National Taiwan University Abstract: It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. Abstract and Applied Analysis (1997) Volume: 2, Issue: 1-2, page 137-161; ISSN: 1085-3375; Access Full Article top Access to full text Full (PDF) How to cite top 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. The isoperimetric inequality for minimal surfaces (see, e.g., Chakerian, Proceedings of the AMS, volume 69, 1978). A classi ca-tion theorem for complete stable minimal surfaces in three-dimensional Riemannian manifolds of nonnegative scalar curvature has been obtained by Fischer-Colbrie and Schoen [3]. Finally, we establish an index estimate and a diameter estimate for free boundary MOTS. Stable minimal surfaces have many important properties. Deutsch. It is obvious that a complete stable minimal hypersurface in \(\mathbb{H}^{n+1}(-1)\) has index 0. We note that the construction of the index in this space (in the sense of Fischer-Colbrie [FC85]) in Section 4 is somewhat subtle. A minimal surface S of general type and of maximal Albanese dimension satisfies the Severi inequality K 2 S ≥ 4χ (K S) (). PubMed Google Scholar, Barbosa, J.L., do Carmo, M. Stability of minimal surfaces and eigenvalues of the laplacian. © 2021 Springer Nature Switzerland AG. [SSY], [CS] and [SS]. The Sobolev inequality (see Chapter 3). So we get the minimal surface equation (MSE): div(ru p 1 + jruj2) We call the solution to this equation is minimal surface. For the minimal surface problem associated with (6) – (7), it is shown in Section 4 of Chapter … For the … %PDF-1.5 Let X be a smooth minimal surface of general type over kand of maximal Albanese dimension. a time symmetric Cauchy surface, then θ+ = 0 if and only if Σ is minimal. of Math.88, 62–105 (1968), Schiffman, M.: The Plateau problem for non-relative minima. inequality to higher codimension, to non local perimeters and to non euclidean settings such as the Gauss space. 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 Barbosa, J. L. (et al.) Theorem 3. Barbosa J.L., Carmo M.. (2012) Stability of Minimal Surfaces and Eigenvalues of the Laplacian. Theorem 3.1 ([27, Theorem 0.2]). § are C1, parameterized by arclength, such that the tangent vector t = c0 is absolutely continuous. can get a stability-free proof of the slope inequality. Barbosa, João Lucas (et al.) This is a preview of subscription content, access via your institution. Rational Mech. Marginally trapped surfaces are of central importance in general relativity, where they play the role of apparent horizons, or quasilocal black hole bound-aries. It became again as a conjecture in [Ca,Re]. More precisely, a minimal surface is stable if there are no directions which can decrease the area; thus, it is a critical point with Morse index zero. Then!2 X=k 4˜(O X): Let us brie y introduce the history of this inequality. 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 Brasil. - 85.214.85.191. xڵ�r۸�=_�>U����:���N�u'��&޶3�}�%��$zI�^�}��)��l'm������@>�����^�'��x��{�gB�\k9����L0���r���]�f?�8����7N�s��? Arch. Mech.14, 1049–1056 (1965), Spruck, J.: Remarks on the stability of minimal submanifolds ofR Curves with weakly bounded curvature Let § be 2-manifold of class C2. Classify minimal surfaces in R3 whose Gauss map is … The The stability inequality (where D is the covariant derivative with respect to the Riemannian metric h) ⑤Dα⑤ 2 … A Reverse Isoperimetric Inequality and Extremal Theorems 3 1. On the size of a stable minimal surface in R 3. A stability criterion can be seen as a set of inequality constraints describing the conditions under which these equalities are preserved. If rankL = 1 or 2 then x(M) is a quotient of the plane, the helicoid or a Scherk's surface. Suppose that M is connected and has finite genus, and suppose that x : M —>T\?/L is a complete, stable minimal immersion. Immediate online access to all issues from 2019. Math Z 173, 13–28 (1980). A Stability Inequality For a Class of Nonlinear Feedback Systems (M114) A.G. Dewey. 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. Department of Mathematics Technical Report19, Lawrence, Kansas: University of Kansas 1968, Chern, S.S., Osserman, R.: Complete minimal surfaces in euclideann-space. The minimal area property of minimal surfaces is characteristic only of a finite patch of the surface with prescribed boundary. Acad. Speaker: Chao Xia (Xiamen University) Title: Stability on … Stable approximations of a minimal surface problem with variational inequalities Nashed, M. Zuhair; Scherzer, Otmar; Abstract. Stable approximations of a minimal surface problem with variational inequalities. Then, the stability inequality reads as R D jr˘j2 +2K˘2 >0. Exercise 6. We do not know the smallest value of a for which A-aK has a positive solution. Hence our theorem can be regarded as an extension of the results in [ 6 – 8 ]. If (M;g) has positive Ricci curvature, then cannot be stable. It was Severi who stated it as a theorem in [Se], whose proof was not correct unfortunately. Index, vision number and stability of complete minimal surfaces. 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. outermost minimal surface is a minimal surface which is not contained entirely inside another minimal surface. Finally, Section 6 gives an account on how the techniques developed for the isoperimetric inequality have been successfully applied to study the stability of other related inequalities. 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. Math. 162, … The case involving both charge and angular momentum has been proved recently in [25]. In fact, all the known proofs are related to some sort of stability: geometric invariant theory in [CH] (in Lemma. Arch. strict stability of , we prove that a neighborhood of it in Mis iso- ... of a stable minimal surface ˆMwas in the proof of the positive mass theorem given by Schoen and Yau [17]. 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. Comm. Again, there is a chosen end of M3, and “contained entirely inside” is defined with respect to this end. First, we prove the inequality for generic dynamical black holes. 68 0 obj Scand.5, 15–20 (1957), Polya, G., Szegö, G.: Isoperimetric inequalities of Mathematical Physics. In [10] do Carmo and Peng gave We link these stability properties with the surface gravity of the horizon and/or to the existence of minimal sections. It is the curvature characteristic of minimal surfaces that is important. Processing of Telemetry Data Generated By Sensors Moving in a Varying Field (M113) D.J. The Zero-Moment Point (ZMP) [1] criterion, namely that Of course the minimal surface will not be stationary for arbitrary changes in the metric. Many papers have been devoted to investigating stability. Mathematische Zeitschrift Subscription will auto renew annually. 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. Destination page number Search scope Search Text Search scope Search Text On the Size of a Stable Minimal Surface in R 3 Pages 115-128. n An. This inequality … volume 173, pages13–28(1980)Cite this article. The conjectured Penrose inequality, proved in the Riemannian case by 2 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. Publication: Abstract and Applied Analysis. This is no longer true for higher codimensional minimal graphs in view of an example of Lawson and Osserman. Stability of surface contacts for humanoid robots: ... issue, as its dimension is minimal (six). Ci. In particular, we consider the space of so-called stable minimal surfaces. [F] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index in three manifolds, Invent. At the same time, Fischer-Colbrie and Schoen [12], independently, showed Anal.45, 194–221 (1972), Lawson, Jr., B.: Lectures on Minimal Submanifolds. Rational Mech. Springer, Berlin, Heidelberg. 1-forms in the stability inequality with even slower decay towards the ends of the minimal surface than those considered previously. Math. We can run the whole minimal model program for the moduli space of Gieseker stable sheaves on P2 via wall crossing in the space of stability conditions. The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). Math. 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. In particular, F(E) F(K) = njKj whenever jEj= jKj. His key idea was to apply the stability inequality[See §1.2] to different well chosen functions. $\endgroup$ – User4966 Nov 21 '14 at 7:12 Rend. Pogorelov [22]). Ann. https://doi.org/10.1007/BF01215521, Over 10 million scientific documents at your fingertips, Not logged in

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