PHP Form Issues

<?php

            if(isset($_POST['submit'])){

                $to = "tiffany@tmcallister.net";
                $subject = "Contact form submission from $name";
                $name = $_POST['name'];
                $email = $_POST['email'];
                $message = $_POST['message'];

                if ($name == "" || $email == "" || $message == ""){
                        $error_blank = "<p>One or more required fields have been left blank. Please try again.</p>";
                }
                elseif (!filter_var($email, FILTER_VALIDATE_EMAIL)){
                        $error_email = "<p>The email address you entered is not valid. Please try again.</p>";
                }
                elseif (mail($to, $subject, $message, "From: $name\n")) {
                        $success = "<p>Thanks for the message! I will get back to you as soon as possible.</p>";
                } 
                else {
                        $error = "<p>Something went wrong and the form was not submitted. Please try again.</p>";
                }
            }
        ?>

        <!-- CONTACT SECTION -->
        <section id="contact">
            <div class="container">
                <h2>Contact</h2>

                <?php if(isset($error_blank)){
                    echo $error_blank;
                }
                elseif (isset($error_email)){
                    echo $error_email;
                }
                elseif (isset($success)){
                    echo $success;
                }
                elseif (isset($error)){
                    echo $error;
                }
                ?>

                <form role="form" action="<?php htmlentities($_SERVER['PHP_SELF']); ?>" method="post">
                    <div class="form-group">
                        <label for="name">Name:</label><span class="label label-danger"><span class="icon-warning"></span> This is a required field.</span>
                        <input class="form-control" id="name" type="text" name="name" value="<?php if (isset($name)) { echo htmlspecialchars($name); } ?>" placeholder="Type your name">
                    </div>
                    <div class="form-group">
                        <label for="email">Email:</label><span class="label label-danger"><span class="icon-warning"></span> This is a required field.</span>
                        <input class="form-control" id="email" type="email" name="email" value="<?php if (isset($email)) { echo htmlspecialchars($email); } ?>" placeholder="Type your email address">
                    </div>
                    <div class="form-group">
                        <label for="message">Message:</label><span class="label label-danger"><span class="icon-warning"></span> This is a required field.</span>
                        <textarea rows="5" class="form-control" id="message" name="message" placeholder="Type your message"><?php if (isset($message)) { echo htmlspecialchars($message); } ?></textarea>
                    </div>
                    <input type="submit" class="btn btn-default" value="Send Message">
                </form>

                <div class="top-arrow">
                    <a href="#page-top" title="Back to top">
                    <span class="icon-up-arrow">
                        <span class="sr-only">Back to top</span>
                    </span>
                    </a>
                </div>
            </div>
        </section>